a. The balloon as a perfect isopycnic tracer

Figure 1 illustrates (in solid) the vertical displacement of an isopycnic surface induced by a gravity wave. We assume here that the balloon always remains on this surface. Let A (altitude z₀) be a point where the wave-induced density disturbance vanishes. At A, the density therefore simply reduces to the background atmospheric density ρ(z₀). Let B be a point where the vertical displacement ζ′_ρ of the isopycnic surface is nonzero, where the prime denotes a wave disturbance. The density at B is

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By definition, it is equal to the density at A so that at first order of the wave disturbance:

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Solving for the vertical displacement of the isopycnic surface yields

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where

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is the density scale height.

The wave-induced pressure perturbation in B is obtained as in (1):

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Hence the pressure perturbation observed by the balloon (i.e., the Lagrangian pressure perturbation) at B is, to first order,

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where the index l denotes Lagrangian disturbances. It should be noted that the second-order Stokes correction to the Lagrangian-mean pressure has been neglected in this equation. The Lagrangian pressure perturbations measured by the drifting balloon are therefore the sum of two components: (i) the Eulerian perturbation caused by the wave p′(z₀), hereafter referred to as p′_w, and (ii) a contribution associated with the vertical gradient of the background pressure ζ′_ρ∂p/∂z, hereafter referred to as p′_g. Similar equations can be written for the velocity perturbations, but we assume in the following that u′_g = ζ′_ρ∂u/∂z and υ′_g = ζ′_ρ∂/∂z are small with respect to the Eulerian perturbations (see, e.g., Hertzog and Vial 2001).

From now on, a complex notation is used for the wave disturbances with primed quantities denoting the complex wave amplitudes. The following developments rely on the classical GW equations derived in Fritts and Alexander (2003). Thus, the horizontal momentum equations can be used to find a relation between the Eulerian pressure disturbance and the velocity perturbation in the direction of propagation of the wave u′_||:

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where δ₋ = 1 − f ²/ω̂² (with f the inertial frequency and ω̂ the intrinsic frequency of the wave) and ĉ is the horizontal intrinsic phase speed of the wave (i.e., ĉ = ω̂/k_h, where k_h is the horizontal wavenumber). The vertical momentum equation [Eq. (17) in Fritts and Alexander (2003)], on the other hand, relates the Eulerian pressure and density perturbations:

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where w′ is the vertical velocity perturbation, and m is the vertical wavenumber. Owing to the sampling period of 15 min used during Vorcore, we cannot resolve the highest frequency part of the gravity wave spectrum. The motions that we are interested in are therefore to a good approximation hydrostatic, and we can safely ignore the first term on the left-hand side of (7).

Assuming that the background pressure is in hydrostatic equilibrium, p′_g can be expressed as

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Combining (3) with (7) and (8), p′_g can be expressed in terms of p′_w, so the Lagrangian pressure perturbation becomes

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Using (6) the Lagrangian pressure perturbation can then be related to the velocity perturbation in the wave direction of propagation:

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It is interesting to note that for a typical GW vertical wavelength of 3 km in the polar lower stratosphere (Allen and Vincent 1995) (i. e., m ∼ 2 × 10⁻³ rad m⁻¹) and with H ∼ 6 km, the imaginary part of (10), which is associated with p′_g, dominates the real part so that p′_l and u′_|| are essentially in quadrature, whereas the Eulerian pressure disturbance is in phase with u′_||.

Forming the covariance of p′_l with u′_||, one obtains

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where the asterisk denotes the complex conjugate. The horizontal intrinsic phase speed of the wave can thus be computed from this covariance:

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where ℜ stands for the real part.

To obtain an expression for the wave absolute momentum flux, we first use the polarization relation between w′ and u′_|| (Fritts and Alexander 2003):

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where N is the buoyancy frequency. Using (11)

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where ℑ stands for the imaginary part. Note that we are only interested in the real part of the momentum flux as it is the only contribution that does not vanish when averaged over a wave period.

Simple calculus shows that the multiplicative term on the right-hand side in the previous equation depends on the vertical gradient of background temperature, so

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where R and c_p are, respectively, the perfect gas constant and specific heat at constant pressure per unit mass of air. Equations (14) and (15) are equivalent to (11) in Hertzog and Vial (2001).

b. Real balloon

In high static stability conditions, such as those encountered in the stratosphere, superpressure balloons behave asymptotically as isopycnic tracers (Massman 1978). Nevertheless, Nastrom (1980) showed that the vertical displacement of a balloon (ζ′_b) can slightly lead that of the isopycnic surface. We now study the effect of this small shift. Let ϕ (>0) be the small angle by which the balloon leads the isopycnic surface (see also Fig. 1):

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The expression for p′_g is modified accordingly:

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while p′_w is unchanged to first order. The resulting new expression for the Lagrangian pressure perturbation becomes

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Now mH ≫ ϕ/2 for typical values of m and H so that, when forming the covariance with u′_||,

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Hence the imaginary part is not modified by the phase shift ϕ, and the wave momentum flux can still be estimated using (14). On the other hand, estimation of the wave intrinsic phase speed now requires knowledge of ϕ. To estimate ϕ from the balloon data, we consider the density recorded by the balloon; namely,

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Using (3) and (16) gives

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to first order. Hence, the Lagrangian density perturbation is

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Using (6) and (7) gives

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Forming the covariance of ρ′_ι with u′_|| and taking the real part yields

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hence, ϕ can be determined via

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Once ϕ is estimated, the velocity perturbation can be computationally shifted to retrieve the intrinsic phase speed of the wave:

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The algorithm used to analyze the long-duration balloon dataset therefore uses (14) and (27) to estimate both the momentum flux and the intrinsic phase speed of the observed GW packets. It ultimately relies on the estimation of phase differences between the observed velocity and pressure time series.

a. Numerical simulations of balloon observations

A three-dimensional model was constructed to simulate the balloon observations. It consists of a background isothermal atmosphere, with a constant buoyancy frequency (N = 2.1 × 10⁻² rad s⁻¹) and a mean eastward flow of 10 m s⁻¹ that is representative of conditions during Vorcore. One or several gravity wave packets, each propagating upward in the atmosphere, were linearly superimposed onto this background. GW polarization relations (Andrews et al. 1987; Fritts and Alexander 2003) were used to compute the wave-induced disturbances in horizontal velocity, pressure, temperature, and density. Wave–wave and wave–mean flow interactions were ignored. GW characteristics, such as the phase constant ψ, intrinsic period 2π/ω̂, azimuth of propagation θ, intrinsic phase speed ĉ, and time corresponding to the center of the wave packet t_c, were chosen randomly within the ranges detailed in Table 1. The wave packets had a Gaussian envelope, with full width at half maximum (FWHM) proportional to the intrinsic period of the wave so that the wave packets always had the same number of oscillations (∼3). Finally, a phase shift (ϕ) between the balloon vertical displacement and that of the isopycnic surface was randomly chosen from a homogeneous distribution between 0 and 0.5 rad and assigned to each wave packet so as to simulate the departure of the balloon from isopycnicity. From the randomly chosen parameters the horizontal and vertical wavenumbers were derived as

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and

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The negative value of the vertical wavenumber means that the waves were propagating upward.

The amplitudes of the wave-induced disturbances in horizontal velocity along the directions of propagation were chosen so as to be consistent with the horizontal kinetic-energy density spectrum displayed in Hertzog and Vial (2001) and Hertzog et al. (2002). In particular, the wave amplitude scales as ω̂⁻¹ in our simulations. The vertical momentum flux transported by the wave packets is computed analytically from the wave parameters and the Gaussian envelope.

For some simulations, the combination of parameters generated unstable waves. In the real atmosphere, such wave-generated instabilities will significantly reduce the wave amplitudes. Simulations that included unstable waves were therefore discarded. To determine whether a wave packet would be unstable, we used a similar threshold to that advocated by Fritts et al. (2006); namely,

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where ζ′ is the amplitude of the wave-induced vertical displacements of air parcels and λ_z is the vertical wavelength. Simulations that contained quasi-inertial wave packets (ω̂ < 1.1f ) or wave disturbances that occurred too close to the beginning or end of the simulated flights (i.e., those for which less than two periods could be observed in the simulated time series) were also removed from further analysis.

The simulated flights had a duration of 10 days and started from 60°S. The simulated observations mimic those of the real balloons. In particular, the same 15-min sampling rate was used, and the horizontal velocities were computed through finite differences of successive balloon positions. The simulations were run with and without an additional Gaussian white noise to assess the importance of measurement uncertainties in the analysis. The standard deviations of the added noise correspond to the accuracy of the sensors used on real balloons—0.25 K for temperature, 1 Pa for pressure and 15 m for the horizontal position provided by absolute GPS measurements. An example of simulated gravity wave disturbances is displayed in Fig. 2. This figure highlights the relatively greater importance of the measurement uncertainties in the pressure records compared to those associated with horizontal velocities. It is also noticeable that the zonal velocity fluctuations lead those in the meridional component, consistent with the fact that the balloons are drifting in the Southern Hemisphere.

b. Retrieval algorithm

In the balloon observations, GWs often appear as individual wave packets. Wavelet analysis techniques were therefore chosen to retrieve GW characteristics since they enable us to account for the nonstationarity of the wave field. Applied to either real or simulated SPB time series, the wavelet technique provides information in time–(intrinsic) frequency space. The algorithm uses a complex Morlet wavelet as the retrieval of wave momentum fluxes and phase speeds is based on phase differences between the various time series. The main steps of the retrieval algorithm are as follows:

1. The time series of pressure, horizontal velocity, temperature, and density are decomposed using wavelet techniques. Morlet wavelets with periods ranging between 1 and 24 h are used. This analysis provides complex coefficients that are representative of the energy and phase of the decomposed signal in small time/(intrinsic) frequency bins. The wavelet coefficients are normalized so that the variance of each decomposed signal is recovered when summed over the whole of the modulus of the squared coefficients (Torrence and Compo 1998). From now on, tildes are used to denote wavelet coefficients. For example, P̃(t_i, ω̂_j) corresponds to the decomposed pressure time series on the jth wavelet, whose central frequency is ω̂_j, in the neighborhood of the ith observation point.

2. To avoid interpreting noise as GWs, wavelet coefficients for which the modulus is smaller than three times the standard deviation of the observational uncertainties are discarded.

3. To determine the horizontal direction of propagation to within an ambiguity of 180°, the horizontal velocity coefficients ũ(t_i, ω̂_j) and υ̃(t_i, ω̂_j) are rotated until the velocity parallel to the rotated direction is maximized. Let ũ_|| and ũ_⊥ be the respective velocity wavelet coefficients in the parallel and perpendicular directions. The 180° ambiguity is resolved by assuming that wave packets propagate upward in the atmosphere, so the absolute momentum flux ℜ(ũ_||w*) obtained from (14) is necessarily positive. Figure 3 illustrates this detection for the wave packet shown in Fig. 2. In this case, the input and retrieved direction of propagation are 356° from east.

4. Each wave packet actually projects onto several wavelets, so the central frequency of a wave packet must be known in order to retrieve the momentum flux [see Eq. (14)]. This is achieved by using the polarization relation that links the parallel to the perpendicular velocity disturbances (Fritts and Alexander 2003). Specifically, the wavelet coefficients that satisfy

   

   

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   to within an assumed uncertainty (typically 50%) are selected as being associated with the wave packet centers. In (31), f_i is the inertial frequency corresponding to the balloon latitude at time t_i. This search is only performed among those frequencies that satisfy ω̂_j > f_i, that is, those corresponding to gravity waves. Note that this algorithm can, in principle, detect several wave packet centers at the same observation time. We use [ω̂^c_k(t_i)]_k to denote the set of wavelet frequencies corresponding to these central coefficients.
5. Next, the envelopes of the wave packets are identified. This is done by searching in the frequency space around each ω̂^c_k(t_i) and selecting the neighboring frequencies for which

   

   

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   In contrast with the previous step, this latter search is not limited to GW frequencies alone, as the wave packet may project on wavelets with frequencies smaller than the local inertial frequency. The wavelet frequencies selected during this step are those associated with a gravity wave packet, hereafter referred to as ω̂^gw(t_i). Figure 4 demonstrates the two last steps for the time series corresponding to the wave packet shown in Fig. 2. The estimated wave envelope is displayed in the left panel in the time–frequency plane, whereas the right panel shows the selected frequencies at one specific time. In this case, with only one wave packet in the simulation, the algorithm obviously succeeds in identifying the wave envelope. Some arguably false detections at periods much longer than the wave central period are nevertheless observed. They are, however, of little significance for the GW retrievals since the momentum fluxes associated with these wavelet coefficients are very small.
6. The contribution of the absolute GW momentum flux at time t_i to the flight-mean absolute momentum flux is computed by summing over the selected frequencies; that is,

   

   

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   The phase speeds, on the other hand, are estimated for each of the wave packets present at time t_i. Equation (27) is used at the central frequency ω̂^c_k(t_i) to compute the phase speed of each wave packet. This phase speed is then assigned to the frequencies representative of the wave packet envelope.

a. Simulations with one gravity wave packet

One thousand time series simulating balloon flights disturbed by one GW packet were numerically generated and analyzed with the algorithms described in the previous section. For each simulation the flight-mean GW momentum flux was retrieved and directly compared to the flight-mean “true” momentum flux computed from the input wave packet parameters. To estimate the accuracy with which the phase speeds and directions of propagation are retrieved, a flight-mean intrinsic phase speed and direction of propagation were also computed through a momentum-flux weighted average over the selected wavelet coefficients. For instance, the retrieved flight-mean intrinsic phase speed is obtained as

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where the inner sums extend over the frequencies associated with a wave packet at time t_i.

Figure 5 shows the retrieved propagation directions with respect to the input values. Clearly, the retrievals do not exhibit any significant bias. On the basis of these Monte Carlo simulations, the standard deviation of the retrievals is 2°. The wave propagation directions are well estimated as they rely only on the velocity observations, which have small uncertainties relative to the amplitude of the wave-induced velocity disturbances, whatever the intrinsic frequencies of the wave packets.

Figure 6 displays the performance of the analysis in retrieving the flight-mean absolute momentum fluxes. In general, the algorithm succeeds in estimating the momentum fluxes, although a slight systematic underestimation is observed. This underestimation results from the way in which the algorithm selects the wavelet coefficients. Overall, in this case with one gravity wave packet in each simulation, the mean and standard deviation of the ratios between the retrieved momentum flux values and the input values are, respectively, 0.86 and 0.13; that is, the momentum fluxes are underestimated by 14% ± 13% (1σ). The right panel of Fig. 6 shows the ratio of the retrieved to the input momentum fluxes versus the intrinsic periods of the wave packets. It is noticeable that the momentum flux underestimation occurs mostly at intrinsic periods of a few hours, as well as at periods very close to the inertial frequency. For small periods, the discrepancy results mainly from the finite sampling rate of the balloon observations. As previously mentioned, the retrieval algorithm is based on the estimation of phase differences between time series, which incurs significant inaccuracy when the sampling rate is comparable to the wave period. At quasi-inertial periods, on the other hand, the underestimation results from the small vertical displacements for these wave packets. In this case, the pressure-induced perturbations (i.e., p′_g) become comparable to the uncertainty in the pressure measurements, causing a deterioration in the momentum flux retrieval. Arguably, when analyzing real data this latter case is less prejudicial than the former situation as the Eliassen–Palm flux is expected to vanish for quasi-inertial waves (Fritts et al. 2006).

Importantly, since the wave directions of propagation and absolute momentum fluxes are both well retrieved, the zonal and meridional momentum fluxes, respectively u′w′ ≡ u′_||w′ cosθ and υ′w′ ≡ u′_||w′ sinθ, which are the quantities of interest in GW drag parameterizations, can be estimated from the balloon observations. Typically, the uncertainty in the estimation of the zonal and meridional momentum fluxes is similar to that of the absolute momentum fluxes (∼10%) as the wave directions of propagation are almost perfectly retrieved.

The retrieval of the horizontal intrinsic phase speeds is displayed in Fig. 7 for simulations both with and without measurement noise. When the observations are assumed to be perfect, the uncertainty in the retrieved intrinsic phase speeds is relatively small, that is, ∼12 m s⁻¹ for input phase speeds greater than 20 m s⁻¹. However, the dispersion of the retrieval increases greatly at low intrinsic phase speeds, in which situation a wave is actually perceived as quasi-stationary by the balloon. Therefore, the balloon does not sample adequately the wave packet and the analysis fails. Wave packets with such low intrinsic phase speeds are expected to strongly interact with the mean flow at the level where they are observed and may therefore be of less importance for the forcing of the flow above the balloon float altitude.

When realistic measurement uncertainties are included in the simulations, the retrieval of intrinsic phase speed is significantly degraded. The retrieved phase speeds are overestimated by 60% on average, or by 20% if only waves with ĉ > 20 m s⁻¹ are considered, and the overall 1σ standard deviation of the retrieval is 35 m s⁻¹. There are two main reasons to this deterioration. First, the phase speeds are obtained from the Eulerian component (p′_w) of the wave-induced pressure disturbances, which is generally an order of magnitude smaller than its Lagrangian counterpart and thus typically comparable to the uncertainty in the pressure measurement. Second, the phase shift ϕ between the balloon vertical displacement and that of the isopycnic surface has to be known to compute the wave phase speed using (27). Now, this estimation relies on the density observations, for which the uncertainties are mostly due to the instrumental noise in the temperature measurement. This significantly impacts the accuracy to which ϕ, and consequently ĉ, can be obtained.

b. Simulations with multiple gravity wave packets

In reality, there can be more than one wave packet occurring at any given time. To study how this situation impacts retrievals, we generated two additional sets of 1000 time series with 2 and 10 wave packets in each simulation, respectively. Figure 8 illustrates the wavelet analysis for one simulation in each of the sets. In the two-GW case shown in that figure, both wave packets are still clearly separated by the analysis. When this happens, the retrievals with the two-wave set are very similar to those obtained with the simulations with one wave packet. It was found that they depart significantly only when the wave packets in the two-wave time series occur less than 12 h apart (not shown). In that case, the momentum-flux retrieval is slightly noisier and the retrieved momentum flux can, on some occasions, be overestimated.

When 10 wave packets are present, some of the wave packets will always overlap in the time/frequency space, as illustrated in Fig. 8. It is consequently more difficult for the retrieval algorithm to detect the envelope corresponding to each of the various wave packets. This results in a further underestimation of the flight-mean momentum flux. Overall, the momentum-flux underestimation estimated from the simulations is thus 14%, 13%, and 23% in the respective cases with 1, 2, and 10 wave packets in the time series. On the other hand, the 1σ uncertainty in the estimation of the momentum fluxes does not exhibit any significant dependence on the number of wave packets in the time series and varies between 11% and 13%.

It is not possible to use the quoted underestimations to correct results obtained with the real balloon flights since we do not know a priori how many wave packets are present at any given moment in the atmosphere. Nevertheless, we show here that a proxy can be used to determine the amount by which momentum fluxes are underestimated. Namely, the “true” flight-average GW horizontal kinetic energy can be easily estimated in the balloon observations by applying a high-pass filter (with a cutoff frequency corresponding to the smallest inertial frequency during the flight) to the horizontal-velocity time series, so as to isolate the GW signal. On the other hand, a “retrieved” flight-average kinetic energy can be computed from the wavelet analysis as

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where the inner sum extends once again over the wavelet coefficients selected as corresponding to GW packets found at the ith observation record. The proxy is then simply computed as the ratio of the retrieved kinetic energy [obtained from (35)] to the true kinetic energy (estimated with the filtered time series). Figure 9 displays the correlation between the momentum-flux underestimation and the kinetic-energy proxy. Whatever the number of wave packets in the simulated time series, the regression lines are very similar, indicating that this kinetic-energy proxy can be used to estimate by how much our algorithm underestimates GW momentum fluxes.

c. Directions of propagation and phase speeds

In contrast with the case with one wave packet in the simulated time series, (34) cannot be used to estimate either the retrieved phase speeds or the retrieved directions of propagation when several wave packets are present. Instead, Fig. 10 compares the input and retrieved distributions of the flight-mean density-weighted absolute momentum fluxes ρ as a function of the directions of propagation. The result is the average over all simulations of the ρ(θ) distributions.

In the simulations with two GW packets (Fig. 10, left panel), the input absolute momentum fluxes are independent of the wave directions of propagation, as highlighted by the input distribution. There is a clear correlation between the input and retrieved distributions, which indicates that the directions of propagation are still correctly estimated by the algorithm when only two wave packets are present. It is also noticeable that the retrieved momentum fluxes are slightly smaller than the input values.

In the simulations with ten GW packets on the other hand, a distinct dependence of momentum flux on the direction of propagation was deliberately imposed (Fig. 10, right panel); namely, the momentum fluxes scale as sin²(θ/2) (i.e., there are larger westward than eastward fluxes). This nonisotropic distribution allows us to test the reliability of the retrieval by comparing input and retrieved propagation directions. The good agreement observed in the right panel of Fig. 10 demonstrates that wave propagation directions are, indeed, reliably retrieved, even in the case of a large number of wave packets.

The distributions of density-weighted zonal momentum fluxes ρ as a function of the zonal phase speeds ĉ_x are displayed in Fig. 11. As for the directions of propagation, this figure displays the average over all the simulations of the ρ(ĉ_x) distributions. In our simulations, the momentum fluxes depend on the intrinsic phase speeds through a ω̂⁻¹ scaling so that the input distributions have more momentum flux at low phase speeds. In the case of the two-wave packet simulations, the retrieved distribution is approximately the same as the input. However, the phase speed retrieval is less accurate at low phase speeds, in agreement with the results obtained with the single-wave simulations. A significant part of the momentum flux is also wrongly attributed to high phase speed waves so that the overall retrieved distribution is somewhat broader than the input distribution. These defects are amplified when 10 wave packets are present in the simulations. In that case, an even more significant amount of momentum flux is attributed to phase speeds larger than the largest phase speed used in the simulations. Nevertheless, the strong anisotropy of the input distribution is correctly reproduced by the retrieval algorithm.