Identification of equatorial waves using Hough function vectors

Equatorial waves are essential features of the tropical atmosphere dynamics, and their diagnosis is relevant for both operational and climate analysis. Here, we present a method for the identification of equatorial waves using the multivariate spatial structures of the eigensolutions of the linearized shallow‐water equations over the sphere, named Hough function vectors. The method does not require any a priori space–time filtering to separate westward‐ and eastward‐moving waves, and it can be applied to instantaneous or daily circulation fields. The wave types (Kelvin, Rossby, mixed Rossby–gravity, inertio‐gravity) are unambiguously identified by the intrinsic dynamical relationship between the horizontal wind components and geopotential fields represented in the multivariate meridional structures of the Hough function vectors. Both free and convectively coupled equatorial waves can be diagnosed. Moreover, different time filterings can be applied a posteriori, depending on the purposes of the analysis. This article is written as a proof of concept, and direct comparisons are made with results of other methods recently reviewed in a work published in this journal.


INTRODUCTION
Equatorial waves (EWs) are essential features of the tropical atmosphere dynamics, and their linear theory is well described in many textbooks (e.g. Vallis, 2017;Webster, 2020). The essential linear theory of EWs results from the pioneering work of Matsuno (1966), who derived, for the first time, the free wave solutions of the linearized shallow-water equations for a homogeneous incompressible fluid with depth h e in the equatorial -plane. Shortly after Matsuno (1966) had demonstrated the theoretical existence of equatorial "trapped" waves, Maruyama (1967) and Wallace and Kousky (1968) reported observations of westward-and eastward-propagating wave disturbances consistent with theoretical patterns for mixed Rossby-gravity waves and Kelvin waves, respectively. Wallace and Kousky (1968) suggested that the Kelvin waves' disturbances they found could play an important role on the Quasi-Biennial Oscillation, according to the forcing mechanism proposed by Lindzen and Holton (1968). The observed wave disturbances, in the lower stratosphere, would correspond to free propagating waves that were initially excited by moist convection processes in the troposphere (Holton, 1972).
The earliest observational studies of large-scale EWs were based on the analysis of upper air station data, using cross-spectrum-analysis techniques. With the advent of satellite imagery, it was realized that the tropospheric tropical waves could be coupled with convection (Chang, 1970;Wallace, 1971). Several studies based on the analysis of satellite imagery were published in the 1970 and 1980 and provided ample evidence on the existence of convectively coupled EWs (CCEWs). However, it was only in the 1990's, when long-period global satellite and operational data analyses became available, that the first climatological studies of CCEWs were published -(see Kiladis et al., 2009, for a review). Takayabu (1994) and Wheeler and Kiladis (1999) applied a spectral space-time analysis to long-time series of regularly gridded data of brightness temperature T b and outgoing long-wave radiation (OLR), respectively. Both studies identified spectral peaks in the wave number-frequency domain that are consistent with the dispersion curves of different types of waves as predicted by Matsuno's theory. The structures of the waves in the motion and temperature fields were obtained by regressing those fields onto the OLR signal filtered by selected wave number-frequency windows around the spectral peaks (Wheeler et al., 2000). Yang et al. (2003) argued that the spectral space-time method may fail the correct identification of EWs because the dispersion curves may be shifted by Doppler effects due to the background winds. In addition, the assumption of the separability of the vertical and horizontal structures, made in the linear theory, may not be valid. Consequently, they proposed that the EWs could be more properly identified by projecting the zonal and meridional wind (u, v) fields and the geopotential field onto the horizontal structures given by the linear EW theory. The method proposed by Yang et al. (2003) consists of three steps. First, a space-time spectral analysis is applied to each variable to separate eastward-and westward-moving waves. Next, each dynamical variable is projected independently onto a selected set of meridional structures of the Matsuno EW solutions: the parabolic cylinder functions (PCFs). The selected meridional structures have an optimal equatorial trapping scale determined by the meridional wind data. Finally, the three filtered dynamical fields are analysed to assess how consistent they are in the representation of an EW structure. This method can be applied to the (u, v, ) fields at any single isobaric level in the atmosphere. Gehne and Kleeman (2012) developed a method for the identification of EWs that largely follows the spectral space-time analyses method used by Takayabu (1994) and Wheeler and Kiladis (1999). The main difference in the Gehne and Kleeman (2012) method is that, before the spectral space-time analyses, the field variables (T b , u, v, , and divergence) are projected on a truncated basis of PCFs with the same trapping scale as that used by Yang et al. (2003). The spectral space-time analyses is then performed on the calculated projection coefficients for each longitude and each time. The separation of the field variables into equatorial symmetric and anti-symmetric components is performed by analysing the projections onto symmetric or anti-symmetric PCFs. Like in Yang et al. (2003), the method used by Gehne and Kleeman (2012) analyses each variable independently and can be applied at any single isobaric level in the atmosphere.
The aforementioned methods follow univariate approaches in the analysis of EWs, with each variable being analysed independently. A multivariate method to identify free EWs and CCEWs has been presented by Zagar et al. (2009), Castanheira andMarques (2015), and Marques and Castanheira (2018). In their method, the (u, v, ) fields are projected simultaneously onto the three-dimensional (3-D) structures of the free normal modes (3-DNMs) of the linearized primitive equations over the sphere. The 3-DNM structures are given by the product of separable vertical and horizontal structures. For each vertical structure function, the horizontal structures are also given by the product of separable zonal waves and meridional structures. The nature (type) of the waves is determined unambiguously by the meridional structures, which are known as the Hough functions. The equatorial confinement of the waves is determined by the choice of the subset of vertical structures to be analysed.
In the method of Zagar et al. (2009), Castanheira and Marques (2015), and Marques and Castanheira (2018), it is not necessary to perform any previous space-time spectral analysis to separate eastward-and westward-moving waves like in Yang et al. (2003). In fact, the method can be applied without any time filtering. However, compared with the other methods, the aforementioned multivariate method requires a larger amount of data as input and has a higher computational cost. Moreover, it assumes the separability of the vertical structures, which may be questionable as argued by Yang et al. (2003). Therefore, it will be useful to relax the method of Zagar et al. (2009) and Castanheira and Marques (2015), by avoiding the projections onto vertical structures, which will allow the analysis to be done at single atmospheric levels. This will simplify the method and prevent possible mixing of the signals of free propagating waves in the stratosphere and the forced CCEWs in the troposphere, which may be present when analysing a subset of vertical structure functions.
A simplified multivariate method that can be used with instantaneous or daily averaged (u, v, ) fields at single atmospheric levels would be useful for practical weather forecast applications, as discussed in Yang et al. (2021). This kind of method was already proposed by Barton and Cai (2017) based on PCFs and the dynamical constraints imposed by the linearized equatorial -plane shallow-water equations. However, the application of the method is not straightforward, with the amplitudes of the waves calculated indirectly using the projections onto the PCFs. Part of the complexity of the method results from the fact that the authors did not take into account that the multivariate structures of the waves are in fact orthogonal (Matsuno, 1966).
In this article, we will present a method for diagnosing EWs based on the multivariate projection of (u, v, ) fields at single atmospheric levels onto the wave solutions of the linearized shallow-water equations on the sphere (Laplace's tidal equations). Besides the method being able to be applied to single isobaric level data, it can be also used with instantaneous or time-filtered fields. It does not require any previous space-time spectral analysis to separate westward-from eastward-moving waves, as in the method of Yang et al. (2003).
Section 2 will revisit some elements of the EW theory that are essential to understand and discuss the proposed method. The data and methodology will be described in Section 3, with examples of the application of the method given in Section 4. Finally, Section 5 will present the conclusions.

Free vertically propagating EWs
The complete set of wave solutions of the shallow-water equations linearized around a motionless basic state in the equatorial -plane were derived for the first time by Matsuno (1966). As briefly described in the previous section, much work about EWs has been guided by Matsuno's theory. However, as shown by Lindzen (1967), higher order Rossby wave solutions in the equatorial -plane become less accurate approximations of the solutions on the sphere. More accurate solutions can be obtained considering the linearized primitive equations over the sphere and using the log pressure vertical coordinate z: where (u, v, w) are the zonal, the meridional, and the vertical wind components, respectively. The horizontal coordinates are the longitude and latitude . The vertical coordinate z is defined as z = −H ln(p∕p s ), with p denoting the air pressure and subscript s a fixed value close to the surface. Constants Ω and a are the angular velocity of the Earth rotation and the Earth radius, respectively. The constant H = RT∕g is the scale height of a layer with mean temperature T, R is the specific gas constant, and g is the mean acceleration of gravity. The reference vertical profile of density is 0 (z) = s exp(−z∕H), and the static stability parameter of the reference basic state with temperature T 0 (z) is given by N 2 = (R∕H) (dT 0 ∕dz + g∕c p ), where c p is the specific heat at constant pressure.
Assuming for simplicity, and to gain some insights from the theory, that the stability parameter N 2 is constant, then the system of Equations 1-4 has solutions of the form that represent waves propagating zonally and vertically, with zonal and vertical wave numbers k and m respectively and the angular frequency . The factor e z∕(2H) compensates the decrease of the basic state density 0 with z for the conservation of the wave energy.
Substituting Equation (5) into Equations 1-4 and eliminatingŵ( ) between the equations resulting from Equations 3 and 4, one obtains a system of equations for the free oscillations of a thin layer of an incompressible homogeneous fluid over the sphere that are the free oscillations of Laplace's tidal equations: − iv + 2Ωû sin + 1 a dd = 0, − î+ gh e a cos where h e is the equivalent depth given by h e = N 2 ∕{g[m 2 + 1∕(4H 2 )]}.
Introducing the dimensionless variables (Swarztrauber and Kasahara, 1985) Equations 6-8 can be written in the form where W = (ũ,ṽ,̃) T and where = √ gh e ∕2aΩ. Defining the inner product of two vectors W 1 and W 2 as where the asterisk represents the conjugate transpose, it can be shown that L is a Hermitian operator (Swarztrauber and Kasahara, 1985). Therefore, the eigenfrequencies are all real, and any two eigenvectors associated with different eigenfrequencies are orthogonal. The dispersion relationship of the waves (k, m) = 2Ω (k, m) is given by the dependence of the eigenfrequencies on parameter and wave number k; and the solutions of Equation (10) correspond to westward-and eastwardtravelling inertio-gravity waves and westward-travelling Rossby waves.
On account of Hough's (Hough, 1898) classical work, the eigenvectors of Equation (10) where n is an index that represents the meridional scale, and (=1, 2, 3) distinguishes between westward-and eastward travelling inertio-gravity modes and Rossby modes. The meridional wind is in quadrature with the zonal wind and the geopotential height.
With the exception of some asymptotic limits for , the solutions of Equation (10) have to be calculated numerically (Longuet-Higgins, 1968, and references therein). Swarztrauber and Kasahara (1985) described one software to compute the Hough harmonic vectors, ( ) exp(ik ), based on their expansion in series of spherical harmonic vectors. Recently, Zagar et al. (2015) and Marques et al. (2020) developed two open-access software packages based on the same scheme of Swarztrauber and Kasahara (1985).

Forced EWs
In the previous subsection, the theory of free vertically propagating waves was presented for the ideal case of a resting basic state with a constant stability parameter N 2 . The free vertical propagation may be approached above the tropopause where convection does not occur. In such a case, the wave anomalies, at a given time, should exhibit slanted structures due to both zonal and vertical propagation. In the troposphere, where the EWs are generally expected to be forced, their vertical structures are selected by the convective processes involved in the forcing (Andrews et al., 1987;Haertel et al., 2008;Kiladis et al., 2009). Again for simplicity and to get some insight from the theory, we follow Kiladis et al. (2009) and Webster (2020, see their section 8.5) by considering that the convective heating rateQ is proportional to the vertical velocity; that is,Q = N 2 w, where is a positive constant. In this case, considering a diabatic forcingQ in the right-hand side of the linearized thermodynamic Equation (4), one obtains Using Equation (14) for the vertical velocity in Equation (3) yields where N 2 does not need to be constant.
Assuming that the vertical dependence is separable from the horizontal and time dependence in Equations 1,2, and 15, we seek solutions of the form Substituting Equation (16) into Equation (15), one obtains . (17) Therefore, the vertical structures must be the solutions of the following vertical structure equation: where −1∕gh e is the separation constant. For a flat surface, the lower boundary condition is (Andrews et al., 1987) dΨ dt Substituting this condition into Equation (14), the lower boundary for Ψ(z) is obtained as Considering that the tropospheric mass does not cross the tropopause, at a constant pressure p t or, equivalently, at a constant pressure height z t , then an appropriate upper boundary condition is Finally, substituting Equation (16) into Equations 1,2, and 15, and using Equation (18), one obtains again a system of shallow-water equations: Equation (18) with the boundary conditions given by Equations 20 and 21) constitutes a Stourm-Liouville problem whose eigensolutions determine the vertical structure of the waves and the respective equivalent depths. The solutions Ψ(z) are real functions, and the forced waves in Equation (16) are horizontal travelling waves (Haertel et al., 2008;Kiladis et al., 2009).
If we consider a dry atmosphere ( = 0) and apply Equation (18) to the whole vertical extent of the atmosphere -that is, if we consider boundary conditions at the surface and the top of the atmosphere -then Equations 18-24 are formally the same as the systems treated by Kasahara and Puri (1981), Zagar et al. (2015), and Marques et al. (2020) to analyse the free oscillation modes of the atmosphere; that is, the free zonal travelling waves (Andrews et al., 1987).
The parameter in Equation (24) implies that, if the waves are coupled with convection, the zonal phase velocity in the moist atmospheric layer can be smaller than the phase velocity in a dry atmosphere. In fact, in this case, we have = [(1 − )gh e ] 1∕2 ∕2aΩ, and, in the case of the equatorial -plane approximation, the zonal velocity of pure gravity waves would be c = [(1 − )gh e ] 1∕2 .

More realistic basic states
In Section 2.1 we considered N 2 constant. However, the theory is also useful in the case of a slowly varying stability parameter N 2 (z). In such a case, we can use the Wentzel-Kramer-Brillioun-Jeffreys approximation and consider that the variation of N 2 introduces refractive effects changing the vertical wavelength (see Kiladis et al., 2009, Table 1). It was also assumed that a motionless basic state exists, but the real atmosphere typically has strong background zonal flows. A first step to approach the real basic state flow is to consider a uniform zonal flow U. In such a case, the results are only affected by a Doppler effect, with the intrinsic frequency being replaced by − kU (Lindzen, 1967;Dias and Kiladis, 2014).
In a further step to approach real basic state flows, Kasahara (1980) and Kasahara (1981) analysed the effect of a latitude-dependent basic zonal flow U( ) on the free oscillation modes on the sphere and showed that the low-order meridional modes are not significantly affected.
For zonally varying basic zonal flows, the zonal dependence of the wave solutions of the linearized primitive equations cannot be separated as in Equation (5). If the spatial scale of wave packets is shorter than the longitudinal scale of variation in the background zonal flow, we can invoke the Wentzel-Kramer-Brillioun-Jeffreys approximation and consider that the wave form in Equation (5) is locally valid. However, the advection of the wave packets by the background zonal flow, and the consequent Doppler shifts in the wave frequencies, will change with longitude as the wave packets propagate. Moreover, in the analysis of Hovmöller diagrams, as done in Section 4, it may be expected that the wave anomalies will not be completely aligned with straight lines because the phase velocity may change with longitude.

METHOD AND DATA
A wave number-frequency power analysis of the (u, v, ) field projected onto a single vertical structure, as in Castanheira and Marques (2015), reveals both free EWs and CCEWs (see their figs 9 and 12). This should be an expected result, because the vertical structures that they used consider the whole vertical extent of the atmosphere with forced waves in the troposphere and free waves in the stratosphere. However, as discussed in Section 2.2, real vertical structure functions represent steady vertical structures of zonally propagating waves and are not appropriate to describe vertically propagating waves (Haertel et al., 2008;Kiladis et al., 2009). Moreover, when analysing single vertical modes we are applying the maximum details from the shallow-water wave theory. Because of the complexity of the background state of the real atmosphere, the dispersion curves will be affected by Doppler effects and the waves may actually not have separable vertical and horizontal structures. Following the idea of Yang et al. (2003), we apply here a methodology for identifying EWs using a minimum of details from the linear shallow-water wave theory over the sphere. We will use only the meridional structures given by Hough vectors k,n,m ( ), Equation (13), to filter the data on single isobaric levels. The method consists of the following steps: 1. A wave number Fourier analysis is applied to the variable u, v, and at each time and each latitude of a given isobaric level, obtaining the variables u k ( , z, t), v k ( , z, t), and k ( , z, t). 2. The variables are made dimensionless according to the variable transformation in Equation (9) using a preselected phase velocity c = (gh e ) 1∕2 to obtain the variables u k ( , z, t),ṽ k ( , z, t), and̃k( , z, t). 3. Next, the dimensionless variables are projected onto the Hough vector of the shallow-water system with the preselected equivalent depth h e = c 2 ∕g. The Hough vectors form an orthonormalized basis of vectors under the inner product where i is the Kronecker delta and c is the parameter that controls the equatorial trapping scale as in Yang et al. (2003) or in Gehne and Kleeman (2012). The projection coefficients W kn (c; z, t) are obtained as 4. Finally, the filtered fields in physical space can be recovered by performing the inverse transforms.
This method does not perform any previous space-time spectral analysis to separate the dynamical fields into eastward-and westward-propagating components as done by Yang et al. (2003). The Hough vectors are orthogonal function vectors and can distinguish mixed Rossby-gravity (MRG) waves, equatorial Rossby (ER) waves, Kelvin waves, and eastward-and westward-travelling inertio-gravity waves.
To illustrate the method, we will analyse the same case studies as recently analysed by Knippertz et al. (2022). The work of Knippertz et al. (2022) compared six different methods for the identification of EWs. Our goal is a direct comparison of our results with the results of those methods. Then, we will apply the aforementioned method to the circulation field of ERA5 reanalysis for the 18-year period of 2001-2018. The u, v, fields were downloaded at 6 hr intervals with 1.5 • × 1.5 • (latitude × longitude) horizontal resolution at various isobaric levels.

Sensitivity of the method to the choice of phase speed c
The phase speed c (=(gh e ) 1∕2 ) in our method determines the equatorial trapping scale of the Hough modes, just as the parameter y 0 in Yang et al. (2003) and Gehne and Kleeman (2012) defines the equatorial trapping scale of the PCFs. It is useful to assess the sensitivity of the method to the choice of the value of c.
Because the circulation field is made dimensionless by the changes of variables, Equation (9), the Hough vectors are also dimensionless. If we consider all components of the circulation field with dimensions of velocity, then by defining a new vector In this case, the orthonormalization condition in Equation (25) becomes and the projection coefficients W kn (c; z, t) are again obtained as The new variable (û,v,̂∕c) emphasizes that, in addition to determining the equatorial trapping scale of the Hough modes, the phase speed c also determines the relative contribution of geopotential for the projections onto the Hough modes.
To assess the sensitivity of the wave filtering to the equatorial trapping scale of the Hough modes, one can calculate the spatial correlation r; that is, the inner product between modes of different equivalent depths but the same ( , k, n) indexes: k,n (c; )̂k ,n (c ′ ; ) cos d .
(32) Figure 1 shows the spatial correlation, Equation (32), for Kelvin, MRG, and ER (n = 1) waves. For each type of wave, we have considered h e = 40 m or c ≈ 20 m⋅s −1 as the reference value and h ′ e varying in the range [5, ∼1.0 × 10 4 m], which corresponds to a phase velocity range of 7.0 < c ′ < 316.9 m⋅s −1 . Very high correlations, above 0.9, are observed for different ranges of equivalent depths, which depend on the mode type. This means that, for such ranges of equivalent depths, the meridional profiles of the waves change very little with the equivalent depth. The main differences between the projection coefficients W kn (c; z, t) may result from the relative weight of geopotential in the left-hand side vector field of Equation (27).
As the equivalent depth increases, the amplitudes of the associated modes at subtropical and middle latitudes also increase; that is, the modes become less equatorially confined. Therefore, the correlations between the modes associated with h e = 40 m and the modes associated with large equivalent depths become smaller.
An equivalent depth of h e = 40 m corresponds to a phase speed c ≈ 20 m⋅s −1 , the same as used by Yang et al. (2003) and Gehne and Kleeman (2012) to define the equatorial trapping scale of the PCFs. Moreover, the phase velocities of the EWs associated with an equivalent depth of h e = 40 m are also in the range of phase velocities of CCEWs diagnosed using 3-DNMs (Castanheira and Marques, 2015). In the next subsection, we will reproduce some of the Hovmöeller diagrams for Kelvin, MRG, and ER (n = 1) waves as shown in Knippertz et al. (2022)

Examples of EWs
To allow a direct comparison with the Hovmöller diagrams presented by Knippertz et al. (2022), in step 4 of our method the wave fields were recovered in the physical space by retaining only the wave numbers k = ±1, … , ±15. (The Hough vector associated with a wave number k is the complex conjugate of the Hough vector associated with the wave number −k.) Furthermore, in the next three subsections, the retrieved fields were filtered by applying a Lanczos filter to the 6-month time series from January 1 to June 30 and retaining only periods between 2 and 30 days (the same period range as used by Knippertz et al. (2022)). The period from January 1 Knippertz et al. (2022). Their fig. 10b,d was constructed using the method of Yang et al. (2003) with an equatorial trapping parameter corresponding to a phase speed c ≈ 20 m⋅s −1 as in our Figure 3a,b, whereas their fig. 10a,c was constructed using the 3-DNM method including all vertical structure functions and, therefore, without imposing any particular equatorial confinement.

MRG waves
Figures 4 and 5 show the Hovmöller diagrams for MRG waves. Figure 4 shows the antisymmetric zonal wind troposphere are aligned with the solid green line, which represents a CCEW event identified in the study of Knippertz et al. (2022). The wind components at 200 hPa in Figures 4 and 5 show also eastward-propagating anomalies over the Western Hemisphere. This may be due to advective effects of the background winds. The amplitude of the meridional wind anomalies of the MRG waves does not reduce with the increase of the equivalent depth (Figure 5a-c)), contrarily to what is observed for the Kelvin waves ( Figure 2) and for the antisymmetric zonal wind component of the MRG waves (Figure 4). This raises the possibility that part of the signal recovered on the meridional wind component of the MRG waves may have an extratropical origin. Further analysis of this possibility will have to be postponed to future studies. Fig. 11e of Knippertz et al. (2022), which is the analogous figure to our Figure 5, does not show eastward-propagating anomalies. This can be explained by the pretreatment of the data in the method used to calculate fig. 11e in Knippertz et al. (2022). In fact, using the method of Yang et al. (2003), the eastward-propagating anomalies were filtered out a priori. Figure 6 shows the 200 hPa MRG wave fields on February 21 and 23, 2009. It can be seen that the strong wave pattern, over the eastern Pacific, is displaced westward on day 23 compared with in day 21. The equivalent depth has a similar effect as for the Kelvin wave anomalies; that is, the MRG anomalies are more equatorially confined for smaller equivalent depths (cf. Figure 6a,b with Figure 6c,d). line -as identified in the OLR field by Waliser et al. (2012) and Knippertz et al. (2022). Figure 7a,b is similar to fig. 13k of Knippertz et al. (2022) that was calculated using the method of Yang et al. (2003). On the other hand, Figure 7c has a more patchy structure than Figure 7a,b, showing anomalies propagating westward and anomalies propagating eastward. The Hough modes used in the calculation of Figure 7c should be more sensitive to the extratropical circulation because they are associated with a higher equivalent depth h e = 160 m and, therefore, less equatorially confined than the Hough modes used in Figure 7a,b. This can explain why the ER waves filtered by the 3-DNM method, as used in Knippertz et al. (2022), are dominated by eastward-propagating zonal wind anomalies (see Knippertz et al., 2022, fig. 13l).
The meridional wind anomalies of the ER waves in the troposphere have higher wave-number structures than the zonal wind anomalies (cf. Figure 7a-c with Figure 8a-c). In the upper troposphere, the meridional wind shows both westward-and eastward-propagating anomalies. The amplitude of the eastward-propagating anomalies increase as the equivalent depth increases (cf. Figure 8a-c), which again suggests the influence of the extratropical circulation. It may be observed that the meridional wind anomalies in the lower troposphere predominantly propagate westward with a speed close to that of the identified convectively coupled ER wave (Waliser et al., 2012;Knippertz et al., 2022). This may indicate that the ER waves are less influenced by Doppler shifts in the lower troposphere than in the upper troposphere due to the weaker background winds in the lower troposphere (see Knippertz et al. (2022)).
The differences between the speeds of the westward propagation of zonal wind anomalies in Figure 7a-c and meridional wind anomalies in Figure 8a-c may be due to the existence of both free EWs and CCEWs in the troposphere (Webster, 2020). In fact, the ER zonal wind seems to be associated with free wave propagation, and the ER meridional wind with a slower westward propagation seems to be associated with a CCEW. The different behaviour of the two wind components may be explained by considering the meridional symmetry of the ER (n = 1) waves. From Figure 1, it can be seen that the ER (n = 1) Hough modes associated with an equivalent depth of ∼400 m still have considerable projections onto the modes associated with an equivalent depth of ∼40 m. From the shallow-water theory, we know that fast travelling dry waves are associated with higher equivalent depths and to less confined equatorial structures. Because the meridional structure of the zonal wind in ER (n = 1) waves is symmetric, the main contribution of the zonal wind for the projections onto the Hough modes comes from the anomalies centred at the Equator. Therefore, the contribution of the zonal wind for the projections onto the ER (n = 1) Hough modes will not be very much affected by the equatorial trapping of the mode. On the other hand, the meridional structure of the meridional wind in ER (n = 1) waves is antisymmetric, and the main contributions of the meridional wind for the projections onto the (n = 1) Rossby Hough modes come from the anomalies centred off the Equator. Using Hough modes associated with small equivalent depths, as used here, may impose sufficient equatorial confinement to reduce the contributions of the meridional wind of free ER waves to the projections onto the Hough vectors. This way, the meridional component of the filtered ER waves will be dominated by the contribution of moist waves.

Time filtering of EWs
In the preceding three subsections we analysed the circulation filtered for time periods between 2 and 30 days. However, the spatial wave filtering proposed here can be applied without any time filtering. To assess the effect of the time filtering, we recalculated Hovmöller diagrams similar to those in the previous subsections but proceeded as follows. First, we calculated daily averages from the 6 hr wave fields projected onto the Hough modes associated with an equivalent depth of h e = 40 m. Next, the seasonal cycle was removed from the daily fields in the period of 2001-2018 by subtracting the first three harmonics of the annual cycle. Finally, the Hovmöller diagrams were constructed by filtering for different time periods. We note that removing the seasonal cycle and performing the time filtering before or after the spatial projection onto the Hough modes should make no difference. Figure 9 shows Hovmöller diagrams of Kelvin and ER (n = 1) waves for February 20 to May 20, 2009, considering non-filtered daily data, data filtered for periods less than 30 days, and data filtered for periods between 30 and 96 days. The latter time window was used by Wheeler and Kiladis (1999) and Waliser et al. (2012) to filter the Madden-Julian oscillation (MJO). We choose to represent the Hovmöller diagrams for non-filtered daily data in Figure 9b,e to capture the features shared with each one of the filtered fields. Figure 9a,d shows the Hovmöller diagrams of data filtered for periods of less than 30 days. Both diagrams are almost identical to respective diagrams calculated from 6 hr data filtered for periods between 2 and 30 days (cf. Figures 2b and 7b). The eastward propagation of the Kelvin anomalies is clear in Figure 9a-c, with the propagation of some anomalies in the high-pass (T < 30 days) data being clearly continued in the diagram of the unfiltered daily data. The low-frequency Kelvin waves (periods between 30 and 96 days, Figure 9c) are F I G U R E 9 Hovmöller diagrams of (a)-(c) Kelvin and (d)-(f) equatorial Rossby (ER; n = 1) waves for February 20 to May 20, 2009. Daily data were filtered for periods less than 30 days (a, d), non-filtered (b, e), and filtered for periods between 30 and 96 days (c, e). Shadings and contours are as in Figures 2 and 7 [Colour figure can be viewed at wileyonlinelibrary.com] dominated by small wave-number structures and propagate eastward at a much lower speed than the waves with periods inferior to 30 days. Although the Hovmöller diagram of the non-filtered ER (n = 1) waves ( Figure 9e) has a more patchy structure, some anomalies show also continuity in the westward propagation (Figure 9d). On the other hand, the low-frequency ER anomalies (Figure 9f)) propagate mostly eastward. The propagation of wave signals is less clear in unfiltered daily fields (Figure 9b,e) because the patterns obtained result from the superposition of wave anomalies of the same type but propagating with very different velocities (Figure 9a,c and 9d,f respectively).
The Kelvin and the ER modes are the dominant modes involved in the MJO activity (Castanheira and Marques, 2021). Therefore, the eastward propagation of the anomalies over the Indian and western Pacific oceans, observed in Figure 9c,f, should be associated with the strong MJO activity as documented by Waliser et al. (2012, see their Fig. 4a) for the same period. A clearer association between the low-frequency wave anomalies and the MJO activity observed in the YOTC research programme is shown in Figure 10, which reproduces the main features of fig. 3 from Waliser et al. (2012) but using the divergence of the wind recovered with the Kelvin and ER (n = 1) waves with wave numbers 1-5, and for the time periods between 30 and 96 days. Figure 10 shows the convergence instead of the divergence to make the comparison with the anomalies of the OLR more direct. Cold colours represent positive divergence in the upper troposphere F I G U R E 10 Hovmöller diagram for the convergence of horizontal wind retrieved with Kelvin waves (KW) and equatorial Rossby (ER; n = 1) waves with wave numbers k = 1, … , 5, in the period from May 1, 2008, to April 30, 2010. The convergence was filtered in time with a Lanczos filter retaining the periods between 30 and 96 days. The shading represents the convergence at 200 hPa, and the solid (dashed) contours represent the convergence at 850 hPa equal to 2 × 10 −2 day −1 (−2 × 10 −2 day −1 ) [Colour figure can be viewed at wileyonlinelibrary.com] associated with convection that must be accompanied by negative anomalies in the ORL. The imprint of the three strongest cases of enhanced convection (i.e., strong convergence near the surface and strong divergence in the upper troposphere) associated with the MJO during the YOTC is clearly observed. These three events occurred in April 2009, November 2009, and January 2010.

CONCLUSIONS
The results described in the previous section clearly show that the Hough function vectors can be used as a multivariate spatial filter to identify different types of EWs on single isobaric levels. The method is easier to use than the 3-DNM mode method (Zagar et al., 2009;Castanheira and Marques, 2015) because it needs a smaller amount of data. Furthermore, the method discards the use of predetermined separable vertical structures, and it is not very sensitive to the equivalent depth parameter used to compute the Hough vectors. Unlike the methods that use univariate projections onto PCFs, no space-time spectral analysis is needed in the method presented here to decompose the original data into eastward-and westward-propagating anomalies, and all types of waves are identified unambiguously. In fact, the wave type is determined by the intrinsic dynamical relationship between the horizontal wind components and geopotential fields represented in the multivariate meridional structures of the Hough vectors. The method identifies both free EWs and CCEWs, which may have different signals in the zonal and meridional wind components of the waves, as is the case with ER (n = 1) waves.
Since the method does not need any a priori time filtering, it may be useful for different applications. For example, it may be useful for the identification of EWs in operational forecasts (Yang et al., 2021), or it may be useful in the seasonal forecasts of MJO events by using a low-pass time filtering.