Gravity waves (GWs) play a crucial role in the dynamics of the earth's atmosphere. These waves couple lower, middle and upper atmospheric layers by transporting and depositing energy and momentum from their sources to great heights. The accurate parameterisation of GW momentum flux is of key importance to general circulation models but requires accurate measurement of GW properties, which has proved challenging. For more than a decade, the nadir-viewing Atmospheric Infrared Sounder (AIRS) aboard NASA's Aqua satellite has made global, two-dimensional (2-D) measurements of stratospheric radiances in which GWs can be detected. However, one problem with current one-dimensional methods for GW analysis of these data is that they can introduce significant unwanted biases. Here, we present a new analysis method that resolves this problem. Our method uses a 2-D Stockwell transform (2DST) to measure GW amplitudes, horizontal wavelengths and directions of propagation using both the along-track and cross-track dimensions simultaneously. We first test our new method and demonstrate that it can accurately measure GW properties in a specified wave field. We then show that by using a new elliptical spectral window in the 2DST, in place of the traditional Gaussian, we can dramatically improve the recovery of wave amplitude over the standard approach. We then use our improved method to measure GW properties and momentum fluxes in AIRS measurements over two regions known to be intense hotspots of GW activity: (i) the Drake Passage/Antarctic Peninsula and (ii) the isolated mountainous island of South Georgia. The significance of our new 2DST method is that it provides more accurate, unbiased and better localised measurements of key GW properties compared to most current methods. The added flexibility offered by the scaling parameter and our new spectral window presented here extend the usefulness of our 2DST method to other areas of geophysical data analysis and beyond.

Gravity waves are a vital component of the atmospheric system. These
propagating mesoscale disturbances can transport energy and momentum from
their source regions to great heights. They thus are a key driving mechanism
in the dynamics of the middle atmosphere through drag and diffusion processes

The accurate parameterisation of unresolved gravity waves in global climate
models (GCMs) has proven to be a long-standing problem in the modelling
community. One example of this is the “cold pole” bias

The Atmospheric Infrared Sounder (AIRS)

The Stockwell transform (S-transform)

Here, we present a new analysis method. AIRS radiance measurements are two-dimensional (2-D) images; thus a gravity wave analysis method using a two-dimensional Stockwell transform (2DST) is a more logical approach. In this study we present a 2DST-based method for the measurement of gravity wave amplitudes, horizontal wavelengths and directions of propagation from AIRS measurements. Our method takes advantage of the spatial–spectral localisation capabilities of the S-transform in both dimensions simultaneously, equally and without bias.

South Georgia and the Antarctic Peninsula, together with the southern tip of
South America, lie in a well-known hotspot of stratospheric gravity wave
activity during austral winter, which has been extensively studied both
observationally

In Sect.

Gravity waves can be detected in AIRS radiance measurements as perturbations
from a background state. Here, we use AIRS Level 1B radiance measurements
from the 667.77 cm

The weighting function of the 667.77 cm

If the vertical wavelength is known, it is possible to correct for this
attenuation by dividing the amplitude by an appropriate rescaling factor

In its analytical form, the 1-D Stockwell transform

For a smoothly varying, continuous and one-dimensional function of time

Typically, the scaling parameter

To compute the S-transform using the form in Eq. (

The standard deviation

In this frequency-domain form, the S-transform is computed for each frequency
voice

The S-transform has a number of desirable characteristics for geophysical
data analysis. Unlike a CWT, the absolute magnitudes of the complex-valued
S-transform coefficients in

One disadvantage to using fast DFT algorithms in an S-transform implementation is the familiar coarse wavelength resolution at low frequencies, a limitation not encountered by the CWT. Since both the S-transform and DFT algorithms are easily extended to higher dimensions, however, the reduced computational expense of a DFT-based S-transform makes this a practical tool for large 2-D data sets. Retention of the wave amplitude information in the S-transform is another key advantage.

The S-transform is easily extended to higher dimensions. For a
two-dimensional image

The Gaussian windowing term in Eq.

This is the 2-D form of the 1-D Gaussian window in Eq. (

The specified wave field

As discussed in Sect.

The 2DST is introduced and well described by

To assess the capabilities of the 2DST, it is logical to first apply it to a two-dimensional specified wave field containing synthetic waves with known characteristics.

We create a specified wave field

We first compute the 2-D DFT

For our purposes, we do not need to evaluate

A useful aspect of our implementation is that, like the CWT, we can compute
the 2DST for any individual or range of permitted wavenumber voices by
applying the appropriate wavenumber-scaled Gaussian windows. Although the
permitted wavenumber voices in the spectral domain are evenly spaced, their
corresponding wavelengths are limited to integer fractions (i.e.

The ability to analyse an image at specific wavenumbers is a desirable aspect in geophysical data analysis, where some a priori information regarding the spectral range of wavenumbers detectable in a given data set can be used to reduce the impact of unphysical, spurious or noisy results in 2DST analysis.

The specified wave field

The 2DST

Figure

A 4-D complex-valued function can be difficult to visualise. A
more useful product might be a series of two-dimensional images, the same
size as the input image, that contain the characteristics of the dominant
wave at each location. In the implementation presented here, we neglect
overlapping waves and identify a single dominant wave for each location in

For each such location, we record the complex coefficient of

The location of the spectral peak in

Thus, in the three images

Figure

By taking the real part of the complex-valued image

The 2DST identifies the different spectral regimes of the specified wave field very well, but the reconstructed wave amplitudes are reduced by comparison to their original values.

We suspect the main reason for the reduced amplitudes relates to the “spreading” of spectral power in the transform. Here, as is often the case for gravity waves in the real world, our simulated waves form small wave packets, where wave amplitude decreases around a central location. Such wave packets are usually represented in the spectral domain as some combination of wavenumber voices, in addition to the dominant wavenumber of each of the packets, in order to accurately describe their spatial properties. This means that the spectral power of a single, non-infinite wave packet can be spread across multiple wavenumber voices. Spectral leakage can further contribute to this effect.

The Gaussian window in the 2DST is equal to 1 at its central location but immediately falls away with increasing radius. This means that any spectral power contained in adjacent wavenumber voices, which is required to fully reconstruct the wave, is reduced. When the inverse DFT is computed, the recovered wave amplitude at this location is thus often diminished.

A further reason for the diminished amplitude recovery in
Fig.

Figure

The horizontal wavelength

The direction of wave propagation

To assess the effectiveness of our spectral analysis of the specified wave
field, we compare the known wavelengths and propagation angles of the
synthetic waves in the test image with the 2DST-measured wavelengths and
propagation angles in Fig.

The use of the Gaussian window in the S-transform has some convenient
mathematical advantages; it is analytically simple and has a definite
integral over an infinite range. However, when it is used for 2DST
analysis an unfortunate side effect of the Gaussian window is the poor
recovery of wave amplitude, discussed in the previous section. Although a
Gaussian is traditionally used, any suitable apodizing function may be used,
so long as its spatial integral is equal to unity

In this section, we introduce a new spectral windowing function for the 2DST.
This new function takes the shape of an ellipse in the wavenumber domain, and
a first-order Bessel function of the first kind

As discussed in Sect.

Illustrative surface plots of the wavenumber-domain (top row) and
spatial-domain (bottom row) forms of the traditional Gaussian

One solution to this problem is to use a window that is an ellipse in the
wavenumber domain. Here we introduce an Elliptic–Bessel window

A key feature of this new window is that, in the wavenumber domain, it does
not immediately decrease with displacement from the central location but
rather has a scalable elliptical region within which the function is equal to
unity. Thus, the window captures a much greater extent of the targeted
spectral peak at

One requirement for any apodizing window used in the Stockwell transform is
that the spatial integral of the function must be equal to unity. This is so
that the spatial integral of the Stockwell transform is equal to the Fourier
transform

The normalisation term

The Elliptic–Bessel window

A short derivation of the function in Eq. (

To recap our notation in this study, we have described two windowing
functions for the 2DST: the traditional Gaussian and the new Elliptic–Bessel
windows, which we denote in the spatial domain as

Figure

The surfaces in Fig.

Likewise, Fig.

Orthographic projection of AIRS brightness temperature perturbations

The spatial-domain form of the Elliptic–Bessel window

In the next section we show that the use of the Elliptic–Bessel window in the
2DST, in place of the traditional Gaussian window, significantly improves
wave amplitude recovery. This is very useful for our analysis of AIRS data in
Sect.

A very convenient aspect of the S-transform is its invertibility. Since we
have shown here that both the traditional Gaussian and new Elliptic–Bessel
windows have spatial integrals equal to unity, the 2DST can be completely
inverted to recover the original 2-D image, whichever of these windows or
real non-zero positive values of the scaling parameter

Unfortunately, to take full advantage of DFT algorithms and the inversion
capability of the 2DST for AIRS data, we must compute the 2DST using all
permitted wavenumber voices in both dimensions. This requires nearly 12 000
inverse DFT calculations for each AIRS granule using the traditional
voice-by-voice implementation described here, the computational load of which
could be quite impractical for large-scale studies. Interpolating AIRS
measurements to a coarser resolution with fewer pixels could be one solution
to reduce computational cost, but this will obviously undersample short
horizontal wavelengths in the data. Faster methods for computing the
S-transform have been developed

Figure

In Fig.

We can reduce the impact of this by decreasing the scaling parameter

One problem remains, however. By decreasing

In the general case, these low-amplitude, small-scale variations are unlikely
to be due to gravity waves with vertical wavelengths visible to AIRS, so
their recovery is something we try to avoid. Furthermore, such wavelengths
are very close to or at the Nyquist limit for these data. Our confidence in
their measurement is thus very low, yet the momentum fluxes they transport
can dominate. We discuss this further in Sect.

For the windowing functions considered, it is clear from Fig.

As discussed in Sect.

As a result, the reconstructed images shown in Fig.

Since we have shown that the 2DST is fully invertible for both the Gaussian
and Elliptic–Bessel windowing approaches (Sect.

A possible quantitative metric to assess the first-order effectiveness of our
2DST analysis in Fig.

It is not impossible that in some rare cases the total variance of the
reconstruction could exceed the total variance of the input image, for
example due to the spatial extent of a wave feature being slightly
overestimated. If the localised spectrum for one pixel is affected a larger-amplitude wave feature in one of its neighbouring pixels, this can result in
subtle artificial “borders” between different wave regimes in the
reconstructions. This is not a limitation of the 2DST itself, but arises in
the somewhat forced extraction of localised gravity wave parameters contained
in the 4-D Stockwell transform object

In this section, we use our 2DST-based method to perform gravity wave
analysis on two-dimensional granules of AIRS radiance measurements, comparing
our analysis to that of previous studies. We use the 2DST to measure gravity
wave amplitudes, horizontal wavelengths, and directions of propagation. We
then use ECMWF-derived wind speeds and the assumption of an orographic wave
source to infer vertical wavelengths and make estimates of gravity wave
momentum flux (the vertical flux of horizontal pseudomomentum) by closely
following the method of

The first AIRS granule selected for our study is granule 32 of 6 September 2003, over South Georgia. The second granule is a 135-pixel swath over the intersection between granules 39 and 40 on 2 August 2010, located over the Antarctic Peninsula and Drake Passage.

One limitation of this method is that the phase difference measurements
required to recover along-track wavenumbers can introduce a strong
cross-track bias in resolved features, since the S-transform is only computed
in the cross-track direction. In addition, waves which occupy only small
regions of the granule in the along-track direction may also be
under-represented in the averaged co-spectrum. Furthermore, selecting no more
than five dominant waves in the averaged co-spectrum implicitly limits the
maximum number of available along-track wavenumber voices to no more than
five for each location on the entire granule. The use of a 2DST is a logical solution to each of these problems. With the
increased convenience of computational power since the study of

Before implementing the 2DST, each granule of brightness temperature
perturbations is interpolated onto a regularly spaced grid with approximately
17.7 and 20.3 km separating adjacent pixels in the along-track and
cross-track directions respectively. In the centre of the AIRS swath, the
resolution of this regularly spaced grid closely matches the spatial
resolution of AIRS, so very little if any information is lost. Towards the
edge of the swath, this grid is finer than the spatial resolution than AIRS,
but the grid points will not exactly match the location of the AIRS
footprints. A useful graphic of typical AIRS footprints can be found in

Orthographic projections of a granule of AIRS brightness temperature
measurements

As a result of using DFT algorithms, the maximum numbers of permitted
wavenumber voices available in the along-track and cross-track directions are
limited to

As Fig.

The results of our 2DST analysis of the selected AIRS granules over South
Georgia and the Antarctic Peninsula are shown in Figs.

In both Figures, panel a shows the brightness temperature perturbation
measurements calculated as described in Sect.

Clear wave-like perturbations are observed in both granules directly over and to the east of the mountain ranges. As in previous work, such clear wave-like perturbations are attributable to gravity waves with a high degree of certainty.

Reconstructed 2DST temperature perturbations

The image

Generally, the agreement between reconstructed wave features in
Fig.

Figures

Figures

In Figs.

In the South Georgia granule (Fig.

Here we make estimates of gravity wave momentum flux for the dominant
wave-like features measured by the 2DST in our selected granules, following
the method of

We also have a

Figures

Towards the south-eastern corner of both granules, mean wind speeds become
quite weak. As they fall below around 40 m s

No further wave amplitude attenuation corrections were applied to regions
where vertical wavelengths are inferred to be below 12 km, since this is
beyond the resolution limit of the weighting function of the
667.77 cm

In the South Georgia granule, peak momentum flux values of more than 500 mPa are associated with a small region of large amplitude and short horizontal wavelength wave features, located toward the south-eastern tip of the island. In the Antarctic Peninsula granule, momentum fluxes of a few hundred millipascals are generally co-located with the clearly visible wave structures in the raw brightness temperature perturbations just downwind of the peninsula.

The key strength of the results presented here is the much-improved spatial–spectral localisation and resolution capabilities provided by full two-dimensional treatment of the AIRS data. Confidence in the accuracy of subsequent measured quantities in our 2DST-based analysis is thus greatly improved over previous 1-D S-transform-based methods. Understandably, the former is more computationally intensive than the latter, and this should be considered if data sets are large or computational resources are limited.

Although the magnitude, direction and distribution of momentum fluxes in both
granules are broadly in line with previous AIRS gravity wave studies in the
region

In the implementation of any spectral image processing, it is important to
strike a balance between accurate measurement of the desired properties and
the spurious interpretation of noise. One of the advantages of AIRS
measurements is the high horizontal resolution of the data. With the close
exception of the Infrared Atmospheric Sounding Interferometer

In Fig.

However, gravity waves with very short horizontal wavelengths, tightly packed
in a region immediately downwind of a mountainous island, are in good
agreement with mountain wave theory. Examples of such waves can be found in,
for example, the modelling studies of

An added complication is introduced as a result of the mountain wave assumption used. Since these waves are only a few pixels across, their directions of propagation are difficult to define, introducing a random element. The component of the mean wind parallel to the horizontal wavenumber vector can thus be very low, which decreases the vertical wavelength estimate, which in turn increases the attenuation correction applied to the observed temperature perturbations. This attenuation correction can increase temperature perturbations by 400 % or more which, since momentum flux is proportional to the square of wave amplitude, can increase our estimate of momentum flux to extremely large values.

Thus, by applying a correction factor for wave amplitude attenuation, very
small-scale low-confidence perturbations can yield extremely high momentum
fluxes which can dominate the momentum budget of the entire granule if the
mountain wave assumption is used, correctly or otherwise. This is evident in
the South Georgia granule studied by

As Figs.

Figure

If these extremely high fluxes correspond to real waves, then their measurement is of crucial importance. However, if such perturbations are simply instrument noise and their fluxes are spurious, then the biases and errors introduced by their inclusion in broader studies could be very large.

Without a priori knowledge of the wave environment, which is most readily
gained by visual inspection of the AIRS measurements, it would be unwise to
include the fluxes from these small-scale perturbations in any automated
analysis. Indeed,

Further work investigating this problem is encouraged. Pre-smoothing granules
so as to exclude these perturbations

In this study, we have applied the 2DST
to granules of AIRS measurements, extracting gravity wave amplitudes,
wavelengths and directions of propagation. Our 2DST method builds upon the
work of

We first define our 2DST implementation and test it on a specified wave field containing synthetic waves with known amplitudes, wavelengths and directions of propagation. We find that the 2DST provides very good spatial representation of the dominant spectral components of the specified wave field, accurately measuring wavelengths and orientations of all the synthetic waves.

Due to the spread of spectral power in the spectral domain and wave
undersampling in the spatial domain, we find that localised wave amplitudes
as measured by the 2DST are reduced by more than a factor of two when the
typical Gaussian windowing function is used in the Stockwell transform. We
compensate for this by decreasing the scaling parameter

Next, we measure gravity wave amplitudes, horizontal wavelengths and directions of propagation in two granules of AIRS measurements over South Georgia and the Drake Passage/Antarctic Peninsula region. Our 2DST method significantly improves two-dimensional representation of the dominant spectral features of the granules over previous 1DST methods. These spectral features are directly measured in both dimensions simultaneously for each location of the granule, without the introduction of potential biases caused by the use of averaged co-spectra. This is a clear advantage over previous methods. Another key advantage of our 2DST method is the ability to visually inspect the quality of our spectral analysis. By taking the real parts of the dominant localised spectral coefficients at each location, a reconstruction of the granule can be created. This can be used to fine-tune the adjustable parameters and provide a useful sanity check on the performance of the 2DST. Future work may involve comparing this output to the original data via a variance argument or similar, such that we can obtain a quantitative measure of the quality of the 2DST analysis for quality control purposes in larger-scale studies.

To conclude, our new 2DST-based gravity wave analysis method for AIRS data makes significant improvements over current methods in several key areas, and we would advocate its use in future work.

In Sect.

In this appendix we demonstrate that the Elliptic–Bessel window is admissible
as an apodizing function in the S-transform. To do this, we must first find
the spatial analogue of the wavenumber-domain ellipse we defined in
Eq. (

The Elliptic–Bessel window is defined in the wavenumber

Since

We then recognise that the exponential term in the transform above can be
replaced with sine and cosine functions as

We can omit the last three

This integral can be further simplified if we switch to polar coordinates
using the substitutions

Next we substitute

Next we recall the integral definition of the zeroth-order Bessel function of
the first kind

We now use a new substitution that

Next we use the standard result

Finally, recalling that

This spatial-domain form of the Elliptic–Bessel window in
Eq. (

Equation (

Now that we have found an analytical expression for the spatial-domain form
of the Elliptic–Bessel window (Eq.

This integral can be simplified if we reintroduce our substitutions

Using the standard result

In this appendix so far, we have found a useful analytical expression for spatial form of the Elliptic–Bessel window presented in this study. We have then shown that its spatial integral is equal to unity and it is thus admissible as an apodizing function in the 2-D Stockwell transform. In other cases, a quick test may be performed on candidate S-transform windowing functions to check whether their spatial integral is unity.

If we take the spatial integral

Neil P. Hindle is funded by a NERC studentship awarded to the University of Bath. Corwin J. Wright and Nicholas J. Mitchell are supported by NERC grant NE/K015117/1. The authors would like to thank the AIRS programme team for many years of hard work producing the data used here and also the handling editor and anonymous reviewers for their very helpful suggestions. Edited by: L. Hoffmann