Convectively coupled Kelvin waves in the troposphere have a vertically propagating component which propagates through the tropical tropopause layer into the stratosphere. In the tropical tropopause layer above the typical top of deep convection, these waves propagate as dry waves. In the stratosphere they contribute to the forcing of the stratospheric quasi-biennial oscillation. Here, we address the challenge to track individual waves in a region where both static stability and background wind rapidly change with a new algorithm that operates in real space and uses the full longitude/height/time information available to reliably identify Kelvin waves. We argue that our algorithm overcomes inherent ambiguities in previously published methods. Specifically, our algorithm cleanly separates wave activity and number of waves, and successfully tracks waves also in regions where background wind reduces wave amplitudes. Applied to ECMWF reanalysis data for the period 1989–2011, we obtain a statistical description of Kelvin wave propagation that shows propagation through the TTL into the stratosphere occurs predominantly over the Indian Ocean and Atlantic.
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 Kelvin waves are eastward propagating equatorially trapped large-scale waves, that are convectively coupled in the troposphere and therefore have a complex vertical structure [Kiladis et al., 2009]. Upon entering the stratosphere, they become equivalent to dry waves as predicted by simple linear theory. The transition from convectively coupled to free traveling waves occurs in the TTL, a relatively thin layer around the tropopause over which many important quantities relevant to Kelvin waves, such as static stability, convective forcing and background wind change strongly [Fueglistaler et al., 2009]. Consequently, reliable tracking of Kelvin waves propagating from the troposphere to the stratosphere is a challenge.
 A plane Kelvin wave in linear theory has the dispersion relation
where k and m are the horizontal and vertical wave numbers and ω is the wave angular frequency [Andrews et al., 1987]. ū is the background wind which is assumed constant in plane linear theory, but in reality has a strong zonally asymmetric structure that varies on subseasonal, seasonal and interannual timescales. By considering energy density, it can be shown that westerly winds reduce the amplitude of the waves, and easterly winds amplify the waves. It can also be shown that waves cannot propagate in regions where |ū| > c, where c is the wave speed in the absence of background wind [Andrews et al., 1987]. However, simple ray-tracing (WKB) approximations for the propagation of Kelvin waves are not strictly valid in the TTL, where strong and rapid variation in zonal wind, and more critically in stability, cause a breakdown of the assumption of a slowly varying background.
 In this paper, we present a method for tracking individual waves as they propagate through the TTL in gridded (in space and time) data. The method is applied to ECMWF ERA Interim [Dee et al., 2011] temperature and wind data from 1989 to 2010. Wheeler and Kiladis  introduced a filtering technique to identify Kelvin waves (henceforth referred to as WK99) that has been widely used to study Kelvin waves also in the TTL [e.g., Suzuki et al., 2010; Ern and Preusse; 2009; Chao et al., 2008]. This filter selects a region of spectral space associated with Kelvin waves and transforms only this region back into real space. However, the WK99 method loses some of the spatial structure of Kelvin wave propagation due to its spectral nature, and is also difficult to apply to the vertical tracking problem, making it unsuitable for tracking and following waves. Here, we present a new filtering technique which aims to solve both these problems, and an algorithm for connecting ‘features’ (i.e. locally identified wave patterns) in the filtered data into individual Kelvin waves. The result of this procedure is a collection of individually tracked Kelvin waves.
 We use this collection of waves to investigate the zonal structure of wave propagation in the TTL. In particular, tracking the waves allow us to decompose wave activity, in this case measured by temperature and/or zonal wind variance, into the number of waves and typical wave amplitude. This approach reveals a more complete picture of wave propagation through the TTL. We shall show that this decomposition cannot be done unless information about waves is linked across multiple levels and locations, making the tracking of waves critical to the results presented.
Section 2 gives a detailed description of the method, with supporting analyses given in Appendices A and B. Section 3 investigates the statistical properties associated with waves identified locally, and shows as a consequence that, with only spatially local information, some important details of wave propagation cannot be deduced correctly. This result holds generally, and applies to some previous studies. Section 4shows that by adding non-local information from the wave tracking algorithm described insection 2, we are able to identify waves robustly and solve the problems discussed in section 3. Section 5 compares results based on our algorithm to previously published results, and section 6 presents new results. Finally, section 7 provides a discussion of results and points to intriguing aspects of Kelvin wave propagation that warrant further analysis.
 The method discussed in this paper is a real-space filter, giving us greater control on real-space artifacts of the filter compared to spectral methods such as theWK99 method. A template is used, which is then correlated with the data at every position, showing where the template fits the data best. This process is explained in section 2.2, after the description of the data (section 2.1). Once the data is filtered in this manner, we can track the waves through the TTL, with the tracking algorithm described in section 2.3. In section 2.4 we confirm that the tracking algorithm used in conjunction with our filtering method follows the group velocity of the waves.
 In this paper,we analyze Kelvin waves in the European Centre for Medium Range Weather Forecasting (ECMWF) Interim Re-analysis (ERA Interim) over the period 1989 to 2011. The ERA Interim re-analysis data used here has a horizontal grid spacing of 1° × 1° and a sampling period of 6 hours. ERA Interim has a reasonably high vertical resolution of approximately 1 km in the TTL, which is sufficient to resolve Kelvin waves, which typically have vertical wavelengths of several kilometers. The ERA Interim model is run on hybrid coordinates, and in the TTL these are very similar to pressure levels. Therefore, with the appropriate choice of pressure level, model levels can be interpolated onto pressure levels in the TTL with small interpolation error. The optimum pressure levels minimizing interpolation error are at approximately 67, 80, 96, 113, 132, 154, 177 and 202 hPa. We primarily focus on Kelvin waves in temperature, but also look at Kelvin waves in zonal wind insection 6.3.
 The raw data is separated into symmetric and antisymmetric components to separate waves with similar propagation characteristics but differing symmetry about the equator, as in Wheeler and Kiladis . Kelvin waves are symmetric in temperature, so we use the average between ±10° in latitude. Yang et al.  use the expected theoretical meridional structure to fit to the width of waves, and their results indicate that this is an appropriate choice of latitude band for studying Kelvin waves.
2.2. Filtering the Data
 The real-space filter is a 2-dimensional wavelet transform, where the underlying data is convolved with a wavelet (localized template wave) identifying regions of the data which are similar to the wavelet. The wavelet must resemble a typical Kelvin wave on each level, and also capture the phase relationship of a typical wave packet between levels (we will show insection 2.4 that this is required for tracking wave propagation between levels).
 We can construct a simple analytical template as a plane wave multiplied with a Gaussian window in time (an example of a Morlet wavelet extended to 2 dimensions). The template, , has the form
for relative longitude x and relative time lag t.K is a constant chosen such that . k and ω, the horizontal wave number and period, are parameters in the template which must be chosen to represent the typical characteristics of Kelvin waves in the TTL. ϕ is the phase of the wave, which we will allow to vary as a function of height to capture the correct phase relationship between vertical levels. γ is a free parameter of the Morlet wavelet, and controls the width of the Gaussian window compared with the wavelength of the wave. When γ is small, the filter becomes more temporally local but the apparent frequency of the filter becomes dominated by the scale of the window and not the wave. The effect γ has on the wavelet can be seen in Figure 1. Taking γ ≈ 2 produces a template with a similar correlation timescale to typical Kelvin wave composites and is sufficient to overcome the problems mentioned above when γ is too small. Therefore, in this study we use the value γ = 2.
 We use a standard optimization algorithm to find the values for k, ω and ϕ which maximize the correlation between the template and a composite wave computed using the data. More information concerning this procedure is given in Appendix A. We find that the optimal values found for k and ω do not change much over the TTL, and that the maximal correlation with the composite that can be achieved is not substantially reduced if k and ω are the same for all levels. This simplifies the fitting process and also gives more consistent templates with very similar properties at all levels. This method gives average values of ω ≈ 0.13 cycles per day and k approximately wave number 2.8. This corresponds to the region in the spectra of Wheeler and Kiladis  with the most Kelvin wave power.
 The templates are truncated in x and t to ±30° and ±4 days. Truncation in x gives good spatial locality while still capturing the propagation and structure of the Kelvin waves. The time period chosen to truncate to includes almost all of the power of the Gaussian window. Figure 2 shows an example of the composite and fitted wavelet, showing that a good representation of the wave propagation is captured by the wavelet.
 The filter is applied to the data by convolution with the normalized template centered at each location. This convolution is equivalent to the slope of the regression between the normalized template and the underlying data, and represents the wave component of the signal at each location.
2.3. Tracking Waves in the Filtered Data
 The entire ERA Interim TTL temperature dataset (see section 2.1) averaged over the ±10° latitude band is filtered using the method described in section 2.2. This produces a field which contains ridges where Kelvin wave-like ‘structures’ (i.e. they project onto the template, but may not always be Kelvin waves) are present in the raw data. To locate and track individual waves in this field, we therefore must detect and follow these ridges. This process is crucial for reliable identification of Kelvin waves.
 A simple way to locate these ‘structures’ is to identify all local maxima (taken in a direction which cuts across the ridge) in the data. The waves are more compact in time compared to space (compared to the resolution of the data), and propagate quickly. Consequently, taking the local maximum in the time direction leads to better results because we are closer to being perpendicular to the ridge, and we define a match point as a local maximum in time in the filtered data. These match points are the basis for tracking the waves.
 By linking the match points we seek to derive information about Kelvin wave propagation; for example the phase speed, and variations therein due to variations in background flow. However, in order to link these matches together we must have an a priori estimate for some of the propagation characteristics in order to filter Kelvin waves from other processes which project weakly onto the template. Due care must be taken that the a prioriconstraints do not bias results, and we found that a semi-empirical approach is needed to define the constraints as loose as possible while still linking only Kelvin waves.
 The method we use for determining horizontal links between match points is illustrated by a schematic in Figure 3, and is similar to that of Suzuki et al. . Match points are linked if they lie within a certain range of the expected arrival time, calculated using average phase speeds. Because match points, by definition, must occur at the 6 hourly sampling times, only a finite number of phase speeds can be resolved. Increasing the longitudinal separation of the match points considered, dλ, gives us more resolved phase speeds. Convectively coupled Kelvin waves typically have horizontal phase speeds of 15 ms−1 to 25 ms−1 (from WK99 power spectrum), and therefore we require dλ = 15° in order to resolve this range. A typical Kelvin wave takes approximately 1 day to travel 15°, which is equivalent to four 6 hour samples. The remaining parameter in the method is the maximum allowed time period away from the expected arrival time, ΔtH, which controls the tightness of our constraint on phase speed. The correspondence between phase speed and ΔtH is outlined in Table 1.
Table 1. The Minimum and Maximum Phase Speeds Accepted for Different Values of ΔtH for an Expected Phase Speed of 19 ms−1 and dλ = 15° (Expected Arrival Time of 1 Day)a
See text for further explanation.
 In the vertical, the method used for determining links is very similar. The phase difference between levels in the template has been chosen such that match points in a vertically propagating wave on two adjacent levels occur at the same time. We connect match points on adjacent levels at the same longitude if they lie within a small time period, ΔtV, of each other, to allow some deviation from the typical phase relationship expressed in the template. ΔtV controls the tightness of the propagation constraint when tracking vertically in an analogous way to ΔtH when tracking horizontally.
 ΔtH and ΔtV are parameters to the tracking algorithm and must be chosen appropriately. If they are set too large, multiple neighboring waves will coalesce giving incorrect wave statistics. However, if they are set too tight, waves with slightly atypical phase relationships and propagation speeds will be unduly discarded, seriously biasing results. Appendix B gives details of how we chose appropriate values for these parameters. We find that ΔtH = 12 hours and ΔtV= 6 hours are the optimal choices for ERA-Interim data.
2.4. Phase Relationship
 The tracking method described in section 2.3 must track waves along the group velocity of the wave, otherwise the path followed by the tracking algorithm will leave the wave packet and terminate prematurely. Here, we show that our algorithm does indeed track along the group velocity for Kelvin waves by showing that the theoretical phase of the wave observed when following the wave along the group velocity gives the phase of the wave templates used.
 The total phase of the wave is defined (generally) as
Following the group velocity,
For the special case of a Kelvin wave, the group velocity is parallel to the phase lines of the wave, so kcgx + mcgz = const [Andrews et al., 1987]. This gives the phase of the wave when traveling with the wave at the group velocity as
Rewriting in terms of z and substituting in the Kelvin wave dispersion relation gives
This confirms that using the phase relationship derived by considering a typical Kelvin wave at a fixed time and longitude will track the wave along its group velocity. This shows that these templates must be used for tracking, and that they provide more than just the convenience of linking in this system. This is a special property of the Kelvin wave group velocity, so this method does not apply generally to other waves or disturbances. Other phase relationships could be derived for different types of wave, but these templates would not be as simple to link vertically.
3. Match Distributions
 In section 2.3, we define a match point as a local maximum in the time direction, and note that there are always some local maxima which do not correspond to waves (for example when some other wave or process projects weakly onto the template.) When considering the distribution of amplitudes associated with these local maxima, we might expect a bimodal or other more complex distribution with separation of maxima corresponding to different processes. Figure 4 shows the distribution of the amplitude of these local maxima normalized by the standard deviation of the underlying filtered data on each level. This figure clearly shows that the distribution is not bimodal or complex, and is remarkably invariant over different levels. The distribution also remains similar when location or season are varied (not shown).
 In this section, this invariance will be explained by the surprisingly simple analytical distribution for this type of statistic, which imposes limitations on how we can separate real Kelvin waves from the other local maxima in the data. The simple nature of the distributions shown in Figure 4 suggests that separating Kelvin waves from other processes by use of a threshold will not yield satisfactory results as there is no bimodality in the distribution.
 The limitations this argument imposes on which statistics we can compute directly from the match points also apply to related problems where local maxima are counted with or without a threshold on the data. The consequences for the validity of published results unaware of this problem should be noted.
3.1. Theoretical Distribution of Local Maxima
 Modeling the filtered data as a 1-dimensional Gaussian process, the distribution of local maxima can be calculated analytically [Rice, 1944]. A Gaussian process is a smooth random signal which is Gaussian and with a statistically stationary standard deviation and mean. Such a signal can be represented as an infinite sum over sine waves with random phases. Without loss of generality, we assume the mean is 0.
 The probability density function, f(x), for local maximum amplitude x, is
where . ϵ, the bandwidth parameter, is the only parameter on top of the standard deviation of the underlying Gaussian process, and represents the width of the power spectrum in frequency space. It is a fairly complex function of the moments of the power spectrum, and full details are given in Rice . At very narrow bandwidth, where ϵ → 0, the distribution simplifies to the Rayleigh distribution, and for white noise, where ϵ = 1, the maxima are distributed normally. For intermediate values, the theoretical pdfs are shown in Figure 5.
3.2. Counting Local Maxima
 A useful and meaningful Kelvin wave statistic is the number of waves passing through a given location. A naive method for calculating this is to count the number of local maxima at each location. For the theoretical continuous Gaussian signal, the number of maxima in a given time is simply related to the dominant timescale of the signal and of the bandwidth parameter, ϵ.As discussed above, the non-Gaussian case is not qualitatively different. Kelvin waves have a well defined timescale independent of position. The bandwidth parameter is primarily controlled by the filter used (because the filter strongly controls the shape of the power spectrum), and so again this is likely to be independent of position. Therefore, the number of maxima detected at each location should be approximately constant, rendering this statistic meaningless. Further, it also suggests that regardless of how we define the matches, the method will always include some false positives, as we know that the number of Kelvin waves passing through different locations is not constant in reality.
 In order to remove some of these false positives we could apply a threshold, x0, to the amplitude of the local maxima, as done in Suzuki et al. . The number, n, of maxima above this threshold would then be proportional to
where f(x) is the probability density function for the local maxima with magnitude x, as given in (7). The standard deviation, σ, non-dimensionalizes the variablex, so , where the function depends only on ϵ.With this non-dimensionalized distribution,(8) becomes
If x0 is fixed (and assuming ϵ is constant), the number of waves is simply a function of σ at each location caused by the σ-dependence in lower bound in the integral, and adds no new information to the problem. If the threshold is chosen to be proportional toσ, then the number of maxima above the threshold will stay independent of position as the lower bound remains constant.
 For example, Suzuki et al.  define a threshold in terms of the layer standard deviation. On a single layer this is equivalent to holding the threshold constant, and so variation within the layer will be directly related to variation in the local standard deviation, as shown earlier in (9). The total number of waves detected on each level is roughly constant because of the relationship between the threshold and layer standard deviation. Consequently, applying such a threshold cannot produce meaningful horizontal or vertical structure independent of wave activity.
3.3. Comparison With Theory
 The distribution of the filtered data is non-Gaussian (clear from visual inspection ofFigure 6). This is because the variance of different subsets of the signal can be very different due to a strong annual cycle (and other interseasonal variation) in the wave energy, meaning that the signal is not statistically stationary. Therefore, it does not match the stationary Gaussian process result given in (7) with the same ϵ value. However, we do expect the distribution to behave qualitatively similarly to the theoretical result because, by breaking down the filtered data into Gaussian subsets, we could express the distribution as a weighted sum of Rice distributions (as given in (7)) with different σ values but similar ϵ values. This is possible because the (annual and interseasonal) changes in σ occur on longer timescales than the typical interval between local minima. Taking seasons individually is sufficient to give very close agreement with the Rice distribution predicted by theory (not shown).
Figure 7 shows that the effects described in section 3.2apply to the data, and that the number of matches with either no threshold or a threshold of zero is almost constant. The curves calculated using a higher threshold clearly reflect the form of the standard deviation. This means that any choice of either the filter or threshold will not result in useful information about the number of waves present in the data. In the next section, we argue that the linking process proposed here adds additional non-local information, which in turn allows reliable identification of Kelvin waves, and the separation of the number of waves from wave activity.
4. Using Propagation to Filter Waves
 In section 3we have shown that it is impossible to extract the genuine Kelvin waves from all the match points which satisfy the local maximum criterion without including additional non-local information. Since with our method we track the wave both vertically and horizontally, this can provide non-local information. We shall show that this is enough to allow separation of waves from erroneous matches, resulting in a set of waves which we are confident in. With this robust set of waves, we are then able to separate the number of waves from the wave activity.
 In our approach, templates are applied to each level separately, and with different phase, so vertical connections which occur by chance should be fairly unlikely. Horizontal connections, on the other hand, are more likely since the templates from two adjacent match points separated by the spacing dλ (15°) have overlapping templates. The finite nature of the template does, however, give independence to well separated longitudes, which is not the case with the WK99 filter.
 Waves which propagate over a large number of levels are unlikely to be composed of match points associated with non-Kelvin wave processes. Consequently, we only consider waves which propagate through the TTL from 202 hPa to 65 hPa (corresponding to 8 levels in ERA-Interim). In this paper, we do not investigate the effect that changing the number of levels has on the number and amplitude of waves. We expect that there is some transitional behavior from the raw number of match points (uniform distribution) when no vertical propagation is required, to the robust set of waves obtained when only including those which propagate over a large distance. The results presented here are not sensitive to changes in the choice of levels over which we require propagation. In particular, reducing the number of levels by removing 1 or 2 levels from the upper or lower boundary of our selection does not have a large impact on the results.
 The Wheeler-Kiladis technique [Wheeler and Kiladis, 1999; Wheeler et al., 2000] is a filtering method which operates in spectral space. The region of spectral space associated with Kelvin waves (or any other equatorial wave) is found by comparing the peaks found in the power spectral density (PSD) to dispersion relations. The filter then selects only spectral components which lie in this region before inverting these remaining components back into real space. Subsequent studies, such as Ern et al.  and Suzuki et al. use the Wheeler-Kiladis technique but widen the region associated with Kelvin waves as they find significant power at faster phase speeds when analyzing dynamical fields in the stratosphere. In this study, we take the equivalent depth range 8 m ≤h ≤ 250 m (very similar to that used by Suzuki et al. , and suitable for use in the TTL), and subsequently will refer to this configuration of the filter as the WK99 filter.
 The full filter can be written as
where DFT and IDFT are the discrete Fourier transform and it's inverse, and IR is the indicator function of the region of spectral space corresponding to Kelvin waves, R. Using the convolution theorem, this filter is equivalent to
where ∗ denotes the convolution operator. A very similar result holds for the cross-correlation due to the similarity between the two operators. Therefore, theWK99 filter can be written as a correlation filter with kernel IDFT[IR], allowing us to compare more directly with our correlation filter.
Figure 8 shows the equivalent correlation filter associated with the WK99 filter. We see that the WK99kernel has large non-local contributions that span a large portion of the domain. We expect that this would contribute to a smoothing of the waves, extending them along their direction of propagation, and leading to an overall smoothing of the power distribution. In contrast, our filter is confined locally to ±30° longitude . A typical real-space template is shown inFigure 8in black contours. We see that the phase speed selected is very similar. The time-scale of the real-space filter is longer because the high frequency ringing artifact in the spectral filter is not present. This can be adjusted, although the wave must be resolved within the filter so we cannot use a much narrower (in longitude) filter.
Figure 9 shows the Kelvin wave temperature variance zonal structure and height profiles. The two are separated for clarity because the vertical structure dominates the zonal structure. The vertical profiles of wave activity for the two methods are very similar, showing that both methods are responding to similar features. The zonal structure is also similar, but with more detail, higher zonal gradients, and also more contrast in the results using the correlation filter. This shows that controlling the width of the kernel does indeed enhance the clarity of the distribution allowing more specific questions of propagation and longitudinal structure to be better addressed.
5.2. Wave Lifetime on a Single Level
 Identification of the location of first appearance and eventual disappearance of waves will provide information about the impact of the background flow on Kelvin waves, and may shed light on the distribution of ‘origins’ of Kelvin waves. The previous section showed that there is substantial zonal structure in Kelvin wave activity. This does not in itself imply anything about the start and end of propagation on each level because wave activity is independent of number of waves passing through each location, but does suggest we might expect spatial structure in start and end points.
 By considering only waves which propagate vertically, as discussed in section 4, we can reduce the number of very short erroneous wave-like structures that we track by incorporating information from other levels. As shown inFigures 10a and 10b, waves travel similar distances to those suggested by visual inspection of the data on the 95 hPa level (similar results apply on other levels), with waves starting over the Atlantic and Africa, and ending over the Maritime Continent and the Western Pacific. These patterns of propagation will be examined more thoroughly in section 6.2, where the vertical structure of propagation will also be investigated.
 These results based on our algorithm may be compared to the following approaches. First, one may be tempted to link the matches only on the specific level of interest (i.e. only the horizontal linking). As described in section 3.2, the number of matches at each location is roughly constant. Every match point is included in a wave (since there are no additional criteria) so in this case the start and end distributions become inversely related to the typical distance waves propagate which pass through the given location. This renders these distributions meaningless and misleading.
 Second, we can perform horizontal tracking with a threshold of 1σ, where σ is the level standard deviation of the filtered data, with results shown in Figures 10c and 10d. This approach is in essence similar to that of Suzuki et al. , and has the major caveat that the thresholding forces the number of detected waves at a specific location closely related to the wave activity at that location, as shown in section 3. The 1σ threshold gives a fairly uniform match point distribution (see Figure 7), leading to uniform start and end point distributions in Figure 10d. compared with Figure 10b, where vertical propagation is included. In conclusion, single level wave start and end point results derived using a threshold (such as those presented by Suzuki et al. ) are seriously biased by both wave activity and typical distance propagated through each longitude.
6.1. Average Wave Distributions
Figure 11ashows the structure of the temperature activity calculated using the entire data period filtered with the real-space filter given insection 2.2. The wave activity has been divided by the zonal mean wave activity on each level to remove the dominant vertical structure associated predominantly with N2 variation. Figure 11a shows waves have the largest activity as they propagate through a window from 0° to approximately 100°E. The transition from high wave activity to low wave activity at approximately 100°E at around 125 hPa lies near the background easterly wind maximum. This transition is more clearly defined in this filtered data than in the WK99-filtered results inFigure 9.
 In the upper troposphere, the largest values of Kelvin wave activity are over the mid-Pacific at about 120°W, with another region of increased activity at approximately 0° longitude. There is less Kelvin wave activity over the maritime continent. A significant part of this variability is the wave amplification caused by vertical shear, but the broad pattern extends further down into the troposphere (not shown) implying that there is variation in the convectively coupled Kelvin wave forcing.
 It is tempting to conclude that the westerly winds in the western hemisphere block the waves, but this, although partly true, is not the full story as we will see in section 6.2. However, we do see a significant shadow cast into the stratosphere principally by the Westerlies associated with the upper branch of the Walker circulation, and the Equatorial easterlies in the TTL over the Maritime continent blocking or inhibiting waves. This pattern matches well with the group velocity of Kelvin waves inferred from their structure.
 We have focused on Kelvin waves in temperature so far in this study, as Kelvin waves are larger in amplitude in temperature compared with other processes at and above 113 hPa, due to the increase in static stability at the tropopause. We also note that temperature is better observed in the TTL, especially since GPS occultation satellites such as COSMIC and CHAMP where introduced [Poli et al., 2010]. However, Kelvin waves also project onto zonal wind, which previous studies [e.g., Suzuki et al., 2010] use. There may be important differences in how Kelvin waves project onto these fields which could bias the results, although one might expect that Kelvin waves project similarly onto both fields as plane Kelvin waves are in equipartition between geopotential and kinetic energy. To check for bias, we apply the same procedures to zonal wind. The average wave activity in zonal wind is shown in Figure 11b, and is similar to the temperature activity, shown in Figure 11a, above approximately 125 hPa. Below this level in the upper troposphere, there is less zonal wind activity compared to temperature over the Indian Ocean and Africa, and more zonal wind activity than temperature activity in the western hemisphere. We will discuss Kelvin waves in zonal wind further in section 6.3, where we will look at the results of the tracking algorithms in both fields.
6.2. Propagation Through the TTL
Figure 12 shows the distribution of paths waves take as they travel through the TTL, using the method described in section 2, with a template generated with an index at 95 hPa, counting only waves which can be tracked through the entire region (202 hPa to 65 hPa). Because we only include waves which pass through the TTL, there will be some bias to periods where propagation through the TTL is most likely. Both the QBO and annual cycle strongly affect the wave propagation (not shown here), and Figure 12has to be understood as a superposition of different wave propagation pathways associated with different background conditions. The temporal variability of preferred pathways and their relation to variations in the background state will be discussed in a follow-up paper. Here, we comment on the climatological mean of pathways in the context of the climatological mean background wind field. The reader is reminded at this point that there is substantial variability in the slowly varying background state, and that the climatological mean background state allows only qualitative interpretation of the results.
Figure 12 shows that the background wind clearly modulates wave amplitude, with waves in region A attenuated by easterly winds. In region B, the waves are amplified by the westerly winds, and also blocked by these winds which appear as a strong barrier to wave propagation over the maritime continent. There are few waves propagating through the TTL over the Pacific in region C, and this signal is still clearly observable at C′ in the lower stratosphere.
Figure 12 further shows a marked difference between the distribution of wave amplitude and the number of waves. The separation between amplitude and number of waves is particularly noticeable in region A of Figure 12a. In this region, the background flow leads to small amplitudes but does not block the waves (there are still substantial numbers of waves in this region, as shown in the color shading in the figure). Hence, the ability of our algorithm to follow waves also in regimes with small amplitudes reveals that over the life cycle of a wave, amplitudes may vary strongly and an algorithm sensitive only to amplitude will not provide an accurate description of Kelvin wave propagation.
6.3. Wave Propagation in Zonal Wind
 In section 6.1 we noted that the Kelvin wave activity in zonal wind is similar to the temperature activity, with greater differences in the upper troposphere below 125 hPa. Here, we look at wave tracking results in zonal wind.
Figure 12bshows the distribution of Kelvin waves in zonal wind. Above 125 hPa, the distribution and amplitude of Kelvin waves in zonal wind are very similar to those in temperature. Below 125 hPa, there are some differences in the waves detected. Over the Indian Ocean-Maritime Continent region, there are fewer waves detected in zonal wind on the lowest level (202 hPa), and conversely over the Eastern Pacific in the Walker circulation there are more waves detected with higher amplitudes penetrating deeper into the TTL. In both regions the number of waves detected is more similar between the two sets of results than the wave amplitude.
 The differences between waves in zonal wind and temperature in the upper troposphere reflect the vertical structure of tropospheric convectively coupled Kelvin waves. Convective forcing and the lower boundary condition give rise to a vertical structure with no equipartition between potential and kinetic energy on each level. Differences in the way Kelvin waves project onto zonal wind and temperature in the Indian Ocean-Maritime Continent region and the eastern Pacific region could originate from differences in the type of convection, different static stability profiles or different background zonal wind profiles in these different regions. Models such asFuchs and Raymond  suggest that static stability and zonal wind can change the vertical structure of tropospheric Kelvin waves, and that changes to convective forcing is a coupled response to the different background conditions.
 The consistency in the number and propagation of waves above 125 hPa despite the differences below this level implies that the set of waves tracked in zonal wind is very similar to those tracked in temperature. In the troposphere, waves typically propagate over sufficiently long longitudinal stretches to be robust to localized regions where they are not detected in either field. This suggests that investigations into Kelvin waves in the upper troposphere should take both fields and changing vertical structure into account when drawing conclusions. In this study, this is less critical because we see a consistent set of waves above approximately 125 hPa in both data-sets.
Figure 12 shows that amplitude varies significantly over the lifetime of the wave. Following group velocity lines, which are aligned with the general pattern of propagation, we see that waves passing through the amplitude minimum over the Atlantic (region A) emerge in the stratosphere with an amplitude similar to the waves passing through the window from 0° to 100°E. This suggests that the waves' attenuation of amplitude during passage of this region is not due to dissipation, and recover their original amplitude once they return to a background regime of smaller zonal wind, and smaller vertical shear of the zonal wind.
 Conversely, assuming that the individual waves propagate in a similar direction as the bulk of the waves, we can interpret the decay of the large amplitudes seen in region B upon entering the stratosphere to suggest dissipation of waves in region B. To quantify this dissipation, further analysis would be required to estimate wave energy density and model how this is expected to change in the absence of dissipation. This argument is in practice difficult to demonstrate as the WKB method (ray-tracing) is not strictly valid in the TTL asN2 changes rapidly, so there is no theoretical conserved quantity which can be considered along lines of group velocity. A better understanding of how wave amplitudes change through the TTL, and of processes which force the waves such as convection and radiative cooling, would make this discussion more quantitative and robust.
 Qualitatively, these comments on dissipation agree well with the parameterized mixing included in the ERA Interim model although, as shown in Flannaghan and Fueglistaler , the distribution of this mixing is extremely sensitive to mixing scheme. Because ERA Interim data incorporates some Kelvin wave resolving measurements in the TTL, especially GPS profiles from 2001 onwards, with a large number of observations from 2007 onwards, we expect that the amplitudes observed are not a model prediction but rather driven by the assimilated data.
 The method described in this paper gives new and useful information about Kelvin wave propagation in the TTL that is not available from analysis of wave activity alone. Specifically, the statistics of Kelvin wave propagation produced by our algorithm bears promise to yield insights into Kelvin wave-MJO interaction and Kelvin wave triggering events from a new angle.
Appendix A:: Constructing Wavelets
 In order to filter the data, wavelets are constructed with the same structure as observed by fitting to a composite wave obtained from the data. This process is outlined in section 2.2 and more detail is given here concerning the construction of the composite, and the optimization method.
 Composites are constructed by regressing ERA Interim re-analysis temperature data onto theWK99-filtered brightness temperature at various longitudes. Brightness temperatures from the Cloud Archive User Service (CLAUS) project were used, and these were obtained from the British Atmospheric Data Centre (BADC). CLAUS brightness temperature is binned onto the same 1° × 1° grid as the ERA-Interim data used. Both CLAUS brightness temperature and ERA Interim data are available from January 1989 to June 2006, and this full range is used when constructing the composites.
 The WK99 filter selects a region of spectral space before inverting back into real space. The region used in this study encompasses a broader range of equivalent depths than that used originally in Wheeler and Kiladis , with 8 m ≤ h ≤ 250 m. This is very similar to the filter used by Suzuki et al. . A brightness temperature index is defined at each longitude as the WK99-filtered data at this longitude. Composites are constructed using regression of the ERA-Interim temperature and wind fields onto these indices, such that the composite is centered on the region of maximal convective forcing (negative brightness temperature index). We see some differences as the longitude used to construct the index is varied, so this analysis is repeated at various different longitudes to investigate any longitudinal variability.
 Given a composite Kelvin wave structure, the wavelet parameters k, ω and ϕ are fitted at each level. The fit is performed by maximizing the pattern correlation between the wavelet and composite. Prior to computing the correlation, a Gaussian window of the same width as that used in the construction of the wavelet (see section 2.2 and (2)) is applied to the composite to remove the bias in ω due to the timescale of the wavelet window compared to the time period over which the composite is computed. The maximization process uses a standard optimization algorithm, scipy.optimize.fmin, to maximize the correlation between the composite and wavelet. The initial conditions for optimization must be reasonably close to the solution for k and ω (set by eye), with ϕ set to zero for the first level considered (in the stratosphere). On subsequent levels, the previous k, ω and ϕ are used as initial conditions. This process converges to the correct matching wavelet for all levels in the TTL. Results show that k and ω do not vary much over the TTL, and therefore we fix these to their average values over the TTL in order to make the templates more consistent between levels. This does not significantly change the correlation between wavelet and composite, which remains very high (typically above 0.8).
 For different Tb index longitudes, there is some variation in ϕ(z) and the values of k and ω selected by the optimization algorithm. The templates must be independent of longitude to allow wave tracking using the simple algorithm presented in this paper, and also to ensure consistency between longitudes. Fixing k and ω to the longitudinal average at all longitudes does not affect the optimal correlation coefficient much. The phase of the wave does vary significantly with longitude. However, the phase difference between adjacent levels is the important quantity for the wave tracking algorithm. Figure A1shows deviations in phase difference between adjacent levels relative to the mean phase difference across all index longitudes. This phase difference is expressed in units of time for clarity, and for ease of comparison with the tracking algorithm. We find that throughout most of the domain, the deviation in phase difference from the mean phase difference is less than 6 hours. At most this difference is 18 hours but this is confined to specific longitude-pressure locations.
 The tracking algorithm is tolerant to phase differences (see Appendix B) of less than approximately 9 hours. The regions where phase difference deviates from the mean by more than this are shown as hatched regions inside black contours in Figure A1. These regions are very localized, and unlikely to affect Kelvin waves which typically are tracked for over 90° on each level.
 All calculations shown in this paper were also done with the composites as wavelets. We find results (not shown) to be very similar to those presented here based on the analytical wavelets. Since the composite wavelets are slightly more difficult to use (they have a less uniform time mean, and are less consistent between levels), we chose to present here results based on the analytical wavelet.
Appendix B:: Optimal Parameters for the Tracking Algorithm
 The tracking algorithm described in section 2.3 has two parameters, ΔtV and ΔtH, controlling the tolerance to variations in the vertical and horizontal structures of the wave respectively. If these parameters are too small, Kelvin waves which have a slightly different structure to the template Kelvin wave (representing a typical Kelvin wave) will not be tracked. If the parameters are too large, non-Kelvin wave features which project weakly onto the template will also be tracked.
 Given all match point times on two adjacent levels or longitudes, we can compute the distribution of all time differences between these sets of match points. Figure B1shows the distribution of time differences less than 2 days. Both distributions are broader than we would expect from Kelvin waves alone. If other non-Kelvin wave processes were clearly separated from Kelvin waves, a bimodal distribution might be expected. However, the figure shows that this is not the case, and more sophisticated metrics are required.
 Visual inspection of waves detected in the data with different degrees of constraint on phase speed shows that often waves are joined together in compound wave objects (CWOs), where multiple Kelvin waves are visible within the larger structure. One such object is shown in Figure B2, where the waves are highlighted. Typically we observe that the overall propagation speed of the CWO is less than the expected Kelvin wave speed, suggesting that they are being joined by a slower disturbance which weakly maps onto the Kelvin wave template we use on each level. In some cases, CWOs may arise from differences in Kelvin wave phase speeds with height as a consequence of changes in background wind with height. In this situation, the free wave eventually separates from the convectively coupled wave in the troposphere. However, in general the phase speed differences are large, pointing to another process superimposed on the Kelvin waves.
 Compound wave objects may be of interest in their own right, but in this paper we focus on Kelvin wave propagation through the TTL, and CWOs are an undesired artifact of the algorithm. However, we can use the number of CWOs identified in the data to constrain the tightness of the constraint on phase speed as follows.
 As a proxy to the number of CWOs, we can consider the number of waves which contain multiple matches at the same location but at different times. Horizontal location is defined here as the longitudinal distance along the wave rather than longitude so as not to identify waves which travel completely around the globe as CWOs. We shall call such waves non-planar as they cannot be expressed in (x,p)-coordinates, wherexis horizontal distance along the wave. This criteria is well defined and computationally simple to test for, and is expected to give a subset of all CWOs. The number of non-planar waves as a function of the phase speed constraint parameters is shown inFigure B3.
 In the limit of ΔtH → ∞, we expect to see all the match points at each level join to form one CWO, and thus the number of non-planar waves should approach 1 independent of ΔtV. As ΔtH → 0 we expect waves to become much shorter, breaking up and reducing the number of CWOs and non-planar waves. Therefore, at some intermediate value the maximum number of non-planar waves is observed.Figure B3shows this, with a rapid growth in non-planar waves followed by a longer decay. This rapid growth indicates that a narrow range of phase speeds is responsible for linking waves into CWOs. We therefore conclude that we must link with tight enough constraints on the phase speed to exclude the range of phase speeds which link waves into CWOs and so lie before this rapid growth in CWO number. Similar arguments hold for ΔtV because the links between waves in a CWO commonly involve several layers.
Figure B4 shows the number of waves that can be tracked from 202 hPa to 54 hPa as a function of ΔtH. The number of waves propagating through the TTL will increase as wave fragments are connected into whole waves, meaning that as we increase ΔtH and ΔtV, the number of such waves will initially increase. As in the case of non-planar waves, at large values of ΔtH and ΔtV, waves will join to form CWOs, reducing the number of such propagating waves. The optimum choice of parameters is therefore around the maximum number of these waves. However, the metric is complicated since at large ΔtV, erroneous match points can be connected to span all levels, which is observed to be the case in Figure B4 for ΔtV ≥ 12 hours.
 Taking both metrics together, we conclude that the optimum lies at ΔtV = 6 hours and ΔtH = 6 or 12 hours. Because we can only test at 6 hour intervals, and because there is so much overlap between Kelvin waves and other processes, objectively judging between the 6 and 12 hours for ΔtH using these metrics is not possible. When comparing the propagation of waves between these two cases, we see that there are less waves when ΔtH = 6 hours in regions where the background wind is strong (not shown). Setting ΔtH = 6 hours gives a narrow range of propagation speeds (see Table 1), and in regions of strong background wind we expect atypical propagation speeds, implying that for ΔtH = 6 hours the range of propagation speeds is too tight. Taking ΔtH = 12 hours reduces this bias without forming CWOs as shown by the analysis of the above metrics and therefore is the optimal value.
 We thank George Kiladis and an anonymous reviewer for their very thorough and helpful comments on the first draft of this paper. We thank ECMWF for providing the ERA-Interim data. The CLAUS brightness temperature data used was obtained from the British Atmospheric Data Centre. TF has been supported by a NERC PhD studentship, with additional support from the EU-funded SHIVA project.