A New Way for Walén Test of Alfvénic Fluctuations in Solar Wind Streams via EEMD

Pure Alfvén waves at 1 au can be effectively identified by a Walén test of plasma and magnetic field observations. In this study, a new method, based on the Ensemble Empirical Mode Decomposition (EEMD) technique, is proposed for analyzing Alfvén fluctuations in two ways: (1) the Walén test relied on a time-dependent frame, which is extracted from the trend of differences between plasma and Alfvén velocity; and (2) a modified Walén test of the reconstructed Alfvénic and plasma velocity fluctuations by EEMD, after removing the high/low-frequency components irrelevant to Alfvénicity. Referencing the three validated methods mentioned in Chao et al., the same four cases in high-speed solar wind steams are tested by the EEMD method, considering Alfvénic parameters such as the Walén slope, the standard deviation ratio, the Alfvén ratio, the normalized cross helicity, the normalized residual energy, and the mean deviation relative to 1. The test results indicate that the first EEMD method can improve most Alfvénic parameters, since a time-varying velocity of the De Hoffmann–Teller frame is obtained from the process of self-adaptive data analysis, especially for complex cases with nonstationary and multiple background frames. The second EEMD method provides a kind of flexible testing to reconstruct the optimal fluctuations satisfied for the Walén relation, which achieves similar or even better results than normal prior methods based on bandpass filtering. We suggest that the EEMD method can be an alternative way to identify large-amplitude Alfvén waves in solar wind streams with single-satellite data.


Introduction
Since Alfvén waves (Alfvén 1947) in interplanetary space were observed by Mariner 2 during their flights to Venus in the 1960s (Barnes et al. 1962), many Alfvén fluctuations (AFs) have been found in the solar atmosphere and solar wind, especially in high-speed streams (De Pontieu et al. 2007;Tomczyk et al. 2007;He et al. 2009;Wang et al. 2012;Yang et al. 2016;Martínez-Sykora et al. 2017). According to the observation statistics of Mariner 5, large-amplitude Alfvén waves account for 30% of the solar wind fluctuations at least (Coleman 1967;Belcher et al. 1969). It is believed that Alfvén waves could contribute to energy transmission and dissipation in phenomena as solar corona heating, solar wind acceleration, and geomagnetic disturbances (Alazraki & Couturier 1971;Wentzel 1974;Jacques 1977;Tsurutani & Gonzalez 1987;Matthaeus et al. 1999;McIntosh et al. 2011;Wang et al. 2014). Generally, Alfvén waves can be recognized by the strong correlation between fluctuations of magnetic field B and plasma velocity V from in situ measurements of the solar wind (Unti & Neugebauer 1968), and most of these fluctuations are considered outward propagating Alfvén waves originating near or at the Sun (Belcher et al. 1971). It is also observed that largeamplitude Alfvén waves (relative to the magnitude of the ambient magnetic field B 0 ) frequently occurred at the trailing edge of high-speed streams, with multiscale fluctuations varying from several minutes to a few hours (Bruno et al. 1985;Mavromichalaki et al. 1988).
Under the hypothesis of constant plasma density, thermal pressure, and magnetic field strength, according to the ideal MHD theory, for Alfvén waves correlated fluctuations of plasma velocity and magnetic field satisfy the Walén relation (Walén 1944;Belcher et al. 1969;Hudson 1971; Barnes & Hollweg 1974;Yang & Chao 2013) const. 1 1 2 0 1 2 Here,V andB are the plasma velocity and magnetic field component perpendicular to both the background magnetic field B 0 and wave vector k, r is the plasma density, and μ 0 is the permeability in vacuum; the "+/−" sign represents waves propagating antiparallel/parallel to B 0 . ξ is the thermal anisotropic parameter given by where  P andP are the thermal pressure parallel and perpendicular to B 0 , respectively. The values of ξ was published in the range from 0.8 to 1.0 considering the anisotropies of protons and electrons (Burlaga 1971). Corrections between 4% and 7% follow from the total anisotropies measured on Helios near 1 au (Marsch & Richter 1984).
If the solar wind plasma is assumed to be incompressible, the magnetic field and velocity perturbations dB and dV are related through (Wu 1995) where V A is the local Alfvén velocity in anisotropic media: (1) and (3) represent the theoretical relation. In practice, the observed fluctuations of B and V represent the temporal variations relative to satellites, which actually include some uncertainties unrelated to Alfvén waves. As mentioned above, the remarkable feature of Alfvén waves is that perturbations of plasma velocity and magnetic field are parallel and proportional to each other. Therefore, when it refers to identify Alfvén waves, the simplified Walén relation is tested by a linear fitting between data of plasma and Alfvén velocities for a certain period of time, as a criterion for the degree of Alfvénicity, the so-called Walén test. Meanwhile, appropriate methods are necessary for processing and analyzing the observation data to eliminate the uncertainties unrelated to Alfvén waves, particularly for single-spacecraft measurements.
The conventional method for Walén tests depends on a stationary De Hoffmann-Teller frame (hereafter referred to as the HT frame), which was originally defined by De Hoffmann & Teller (1950) while analyzing the conditions across a onedimensional MHD shock boundary. If a real HT frame exists, or as a reasonable approximation, there is a quasi-stationary pattern, where the magnetic field and the plasma velocity are highly coherent, such as a wave or a current layer. That is, the plasma flows through the shock wave or current layer in the field direction, and the normal component of the HT frame velocity represents the motion of an entire layered structure (Paschmann & Daly 2000). Since the observed plasma velocity V in the instrument frame are composed of the HT frame velocity V HT , the plasma velocity in HT frame ¢ V , and other uncertainty V u (Chao et al. 2014), the HT analysis is an assessment of the uncertainty margins of the HT velocity vector using the minimization procedure. To approximate the V HT from measurements of B and V in a concerned time interval, the common method is to minimize the residual electric field by the minimum variance analysis (MVA; Sonnerup & Cahill 1967), the so-called MVA method. That is, we calculate the convective electric field E (=-V B) using the magnetic field and the plasma velocity in the HT frame, namely In principle, any non-negative merit function can be used to find the best fit for V HT evolution. By solving the conditions ¶ ¶ = V D 0 HT , we can get V HT when D (or E) is minimum, where D is defined as Equation (5) and the superscript t is used to denote the T individual data points in the operation å = T t T 1 1 (Sonnerup et al. 1987). The Walén slope is defined as the ratio of ¢ V V i i A using a least-squares fitting technique (Khrabrov & Sonnerup 1998). Here ¢ V i and V Ai are the ith plasma and Alfvén velocity component (i=x, y, z) in the HT frame, respectively. When the Walén slope is close to 1, with a strong correlation (correlation coefficient ≈1) between plasma velocity and magnetic field, a pure Alfvén wave is qualified. The Walén relation can be simplified as In an HT frame, if the magnetic field structures sampled are stationary, the motional electric field vanishes, and purely Alfvénic fluctuations qualify the Walén test well. However, sometimes the Walén slope is quite sensitive to the concerned time period because the MVA technique depends on sampled data segments with an unacceptable relative difference (Jensen et al. 2013).
As an alternative method, one can estimate the HT frame using another method, in which each component of V HT is calculated from average deviations of the components of solar wind plasma velocity V i and Alfvén velocity V Ai as Equation (7) (Chao et al. 2014), so we call it the mean deviation method. The slopes of x, y, z components are likely to be quite different, so it is considered less credible to evaluate the Walén relation depending on an average of the three component slopes.
Similar V HT and averaged slopes were obtained from the above two methods for the four cases in Chao et al. (2014), identifying the pure large-amplitude Alfvén waves in stationary HT frames. However, it is difficult to find an HT frame when some time-independent structures exist, particularly as a perpendicular structure with a strong intrinsic electric field (Paschmann & Daly 2000). The slopes from the conventional methods are not always credible because the HT frame may change in practice, such as in nonuniform high-speed streams occupied by multiscale dynamic structures. In such cases, the electric field in the constant HT frame does not vanish and the Walén test in this time segment probably provides an inaccurate indication for Alfvénic fluctuations. Likewise, if the HT frame velocity obtained by the mean deviation method does not coincide with the stationary structure, the assumption of Alfvénic fluctuations in all components relative to average values needs to be reexamined (Gosling et al. 2009). Thus, a practical reference frame is expected from the observed plasma velocity to qualify the AFs containing the motion of current layers, discontinuities or any other nonstationary structures.
Considering the uncertainty from the HT frames, Chao et al. (2014) proposed a modified method (hereafter the CHYL method), to check the Walén relation using time difference data between the neighborhood values of plasma and Alfvén velocities, i.e., D = - A . Specifically, when the sampling rate is high enough, the time difference data are independent of V HT and V u , which means Note. The parameters are listed in the following format: observation date (yyyymmdd), start time (hh:mm), end time (hh:mm). These events are selected from Chao et al. (2014). these velocities can be assumed as constant during the period, so Then the Walén relation can be modified as Because time difference data can magnify the values of noises derived from the high-frequency components of an original observation, the values of plasma and Alfvén velocities need to be smoothed by a low-pass filter before generating the difference data in the modified Walén test. This implies that we must preset the frequency range for a bandpass filter without knowing whether the filtered part has an Alfvénic nature. The advantage of this method is that it provides a way to test the Walén relation without relying on any HT frames. But it reduced the original correlation between the plasma velocity V and Alfvénic velocity V A more or less, as seen in the actual results of Chao et al. (2014). Afterward, a new approach was proposed in Li et al. (2016) to check the Walén relation using the multiple-bandpass filtered data instead of the difference data sets.
Considering the multiscale and nonstationary properties of the background streams, a self-adaptive decomposing technique, the Ensemble Empirical Mode Decomposition (EEMD) is applied for the Walén test of Alfvénic fluctuations, and is designed for auxiliary analysis of nonstationary and nonlinear fluctuations. The EEMD technique has recently been applied for numerical and image data, including timescale separation in the solar wind-magnetosphere coupling (Alberti et al. 2017), spectral properties of solar wind density fluctuations (Carbone et al. 2018), temperature variation of sea surfaces (Carbone et al. 2016b), and geophysical turbulences in oceanic environments (Carbone et al. 2016a). In this paper, we find that EEMD can help us remove the irrelevant Alfvénic components from the observation and improve the performance of the Walén test.   This paper is organized as follows. In Section 2, the decomposition procedures of EEMD and the Alfvénic parameters used in this work are introduced. In Section 3, four Alfvén wave events observed by WIND are tested by the EEMD method and other methods for comparison, considering the Alfvénic parameters described in the Section 2.2. Sections 4 and 5 are the discussion and conclusion, respectively.

EEMD
The time-frequency data analysis technique known as Hilbert-Huang Transform (HHT) was proposed by Huang et al. (1998a), and is widely used to process various nonlinear and nonstationary data. It consists of two parts: Empirical Mode Decomposition (EMD) and Hilbert spectral analysis (HSA; Huang et al. 2008).
Unlike traditional time-frequency methods like the Fourier transform and wavelets, HHT is not based on a priori harmonic functions. Instead, it is based on the essential characteristics of the data without setting any basis function. We can use EMD to decompose complex signals into a series of Intrinsic Mode Functions (IMFs) and a monotonous residual naturally (Huang et al. 1998b). The self-adaptive decomposition ensures that the IMFs retain enough physical essence like the amplitude and frequency of the original signals. However, there are some defects of EMD, such as mode mixing, a lack of physical uniqueness and robustness when it is applied to noisy data. As an extended version of EMD, EEMD is a noise-assisted data analysis method developed by Wu & Huang (2004) to overcome the above problems.
The data processing by EEMD is carried out as follows: first, finite-amplitude white noises are added to the original signal as a whole; second, decomposed signals with added white noises go into IMFs according to EMD; then, repeat the above processes for many trials with different random white noises each time; finally, obtain the (ensemble) means of corresponding IMFs. On account of the statistical characteristics of the Gaussian white noise mean value of zero, the final IMFs avoid the influence of adding noises through multiple sets of average processing. Taking the time-series signal as an example, V i (t) represents a component of plasma or Alfvén velocity, which   can be decomposed in terms of a group of IMFs, i.e., where R n is the residue of V i (t) after n IMFs being extracted, the C k (t) is the kth IMF defined by EEMD as follows: As the ensemble number N tends to be infinite, is the portion of the lth trial in the kth IMF in the noise-added signal, with amplitude percentage α as the added noise in each ensemble member versus the original data. In the EMD procedure, to keep the program from going on infinitely, there is a normal criterion for the EMD process to stop: the standard deviation (SD) is computed from the two consecutive sifting results (Huang et al. 2008). The SD is usually set to the value between 0.1 and 0.3, with the same value as α in the procedure.
In this study, EEMD is applied for the Walén test in two ways. The first EEMD method is to extract the HT frame velocity from the decomposed results of plasma and Alfvén velocity difference - The second EEMD method is to test Walén slopes using the reconstructed fluctuations of V and V A , which are combinations of IMFs decomposed by EEMD.

Alfvénic Parameters
In different forms, such as Equations (1), (2), (6), and (8), the Walén relation is always the primary criterion for Alfvénic fluctuations. In addition, there are several approved Alfvénic parameters that can be used to estimate if an Alfvén wave exists or the degree of Alfvénicity.
(1) The correlation coefficient g c and the slope of ¢ V versus V A are primary parameters for identifying Alfvén waves in solar wind. Usually, a linear regression through the origin is recommended to meet the Walén slope (Eisenhauer 2003).
(2) The ratios of SDs of all the components of plasma velocity to Alfvén velocity fluctuations are calculated as s s Chao et al. 2014), reflecting the relative SDs of velocity and magnetic field signals. The correlation coefficient corrected by the ratio of standard variance is equal The Alfvén ratio is defined as This indicates the ratio between the kinetic energy density in the plasma frame and the magnetic energy density of the plasma fluctuations (Tu & Marsch 1995). In Equation (11), dV is the plasma velocity fluctuation vector estimated as and dV A is the Alfvén velocity fluctuation vector estimated as where the bracket denotes the average on the time domain.
(4) The normalized cross helicity is σ c is defined by Matthaeus & Goldstein (1982) as If the above parameters are close to 1 or −1, it is believed that a pure Alfvén wave exists in the plasma velocity and magnetic field fluctuations in nature. ( As a stricter parameter, Li et al. (2016) defines E rr , the mean of SD ratios relative to 1 together with the correlation coefficient between plasma velocity and Alfvén velocity fluctuation: In Equations (13) and (14), the closer σ R and E rr are to zero, the better the Walén relations are satisfied.

Alfvén Waves Cases
Observations of magnetic field and plasma velocities are necessary to search for Alfvén wave events in interplanetary

Walén Test Based on an HT Frame
In this section, three methods are used to evaluate the HT frame velocity and test the conventional Walén relation: the MVA method (Method 1), the mean deviation method (Method 2) and the first EEMD method (Method E1), respectively. For Method 1, the HT frame velocity is satisfied with a minimal residual electric field. For Method 2, the averages of differences between the Alfvén and the plasma velocity components are assumed to be the velocity of the HT frame, as given by Equation (7). For Method E1, the EEMD as a filter allows us to find a nonharmonic trend (residue of EEMD) from the differences between V i and V Ai that is evaluated as the HT frame velocity. In the process of decomposing the velocity deviation (V i −V Ai ) by EEMD, we get a group of IMFs from each component of velocity, and each IMF is the ensemble average of 100 trials (N=100). In each trial, a random white noise is added with the SD of 0.2 (α=0.2). The decomposition results of the above 4 cases are shown in Figure 1, corresponding to curves of original velocity component, all IMFs, and the residue R, respectively. As seen in Figure 1, the number of IMFs obtained from each case is different, C 1 ∼C 8 for Case 1 and Case 3, C 1 ∼C 9 for Case 2, and C 1 ∼C 10 for Case 4, which depends on log 2(T) (Flandrin et al. 2004), where T is the number of data points of original signals.
Case 1 is first taken to demonstrate the test results from the three different methods. By a test  Figure 2(c), which is close to the value derived from Method 1. The linear regression fitting equation of electric field E c versus E HT by method E1 shows a error (1.670) as seen in Figure 2(d) that is slightly larger than method 1 (0.736) as seen in Figure 2(a). This means the residual electric field from method E1 is not the minimum, but the results of the Walén test are reasonable as found by comparing Method 1 and Method 2. Differing from the above two methods, Method E1 provides a time-varying velocity of the HT frame, which changes slowly during the period, with a quasi-monotonic pattern as shown in Figure 2(e). The three components of V HT in GSE coordinates manifest as the ambient solar wind moves through the layer structure; generally, the components have a declining trend, as seen in Figure 2(e). Figure 2(f) shows that both the Walén slope and the correlation coefficient are closer to 1, with values of 1.002 and 0.994, respectively, which are slightly better than method 1. Meanwhile, the slopes of the x and z components are close to 1, but the y component is a little small at 0.744, which shows the poor correlation of V y and V Ay relative to the x and z components. That is possibly caused by some complex structures in the solar wind stream, with nonmonotonic variation of V HTy . This implies that the velocity components show different degrees of coincidence for the Walén relation. Additionally, other Alfvénic parameters are shown in the top left corner in Figure 2(b), (c) and (f), including the Alfvén ratio g A , the normalized cross helicity s c , the normalized residual energy σ R, and the mean deviation E rr . These parameters from Method E1 are all improved relative to Method 1 and Method 2, with γ A (from 1.338 to 1.158) and σ c (from 0.851 to 0.882) changing closer to 1, and σ R (from 0.145 to 0.073) and E rr (from 0.253 to 0.235) changing closer to 0. In Table 2 Tables 3-5 with the same format as Case 1, respectively. For Case 2, the Alfvénic parameters are more accurate than those of Case 1 overall. Significant Alfvénicity is embodied in the strong correlation (γ c 0.95) between ¢ V and V A , as well as the slope and ratio of deviations close to 1. All of the Alfvénic parameters are optimized slightly by Method E1, namely g A , σ c , σ R , and E rr , compared to the results of Method 1 and Method 2. The three components of a time-varying velocity of an HT frame are evaluated by Method E1 with each monotonic trend (an upward trend for V HTx and V HTy , and a downward trend for V HTz , as shown in Figure 3(e)).
For Case 3, the Walén relation is not satisfied very well (the Walén slope is 0.672) though γ c is high, up to 0.986 in Method 1. That means the conditions of the solar wind stream are complicated and it may be more appropriate to analyze it with a non-constant HT frame. Thus, the Walén tests are conducted in a time-varying HT frame (as shown in Figure 4(e)) determined by Method E1 and produce the scatter diagram seen in Figure 4(f). In Table 4, the ratio s s obtained by Method E1 is optimized from 0.682 up to 0.990, and the Walén slope is optimized from 0.672 up to 0.984. It is clear that, comparing with Method 1 and Method 2, in Method E1 it is preferable to satisfy ¢ V and V A with the Walén relation. For Case 4, the velocity and magnetic fluctuations correlate quite well for a long interval over 2 hr, with a γ c of 0.996 and Walén slope of 0.912 for Method 1, as seen in Figure 5(a). Similar to the results for Case 2, using method E1 a monotonous HT frame is found that changes slowly with time, as seen in Figure 5(e). In the time-dependent HT frame, ¢ V and V A satisfy the Walén relation, with a slope of 0.976 and ratio s s of 0.980. Other parameters, including γ A , σ c , σ R , and E rr , are close to values from Method 1 and Method 2, as seen in Figure 5(f) and Table 5.

Walén Test on Difference Data
Without considering the HT frame, two methods can be applied for the modified Walén test, the CHYL method (Method 3) and the second EEMD method (Method E2). According to Chao et al. (2014), for Method 3, five-point smoothing is used to eliminate the high-frequency noises as a low-pass filter before generating the time difference data DV and DV A . In Method E2, we run the EEMD program to deal with components of V and V A for the same four cases mentioned in Section 3.1. First, setting N=100 and α=0.2. The IMFs of each component decomposed by EEMD are plotted from top (high-frequency) to bottom (low-frequency) in Figure 6. Among them, the high-frequency components (e.g., C C , 1 2 ) are mixed with noises from the original data or the random white noises of the EEMD process, while the low-frequency components (e.g., probably contain the variation of background field and the HT frame. It can be seen that each IMF of V i and V Ai corresponds in phase at a very high degree of correlation, especially for the medium-frequency IMFs. The next step is to reconstruct velocity fluctuations by removing some high or lowfrequency irrelevance IMFs (residue). Here, in order to determine the IMF combination, we made lots of attempts, such as~¼C R C R C R , , , n is the most optimal combination; according to the Walén slope it is closest to 1. In this process, we treat the reconstructed IMF components (their sum) as Alfvénic fluctuations for a modified Walén test without considering highfrequency noises and the variation of background fields. The approach with EEMD, just like an adaptive filter, effectively reduces the influences of Alfvénic irrelevant components and the uncertainty from the time-dependent HT frame.
In the results for Case 1, the amplitudes of fluctuations, DV and DV A , are plotted and fitted with linear regression equations in Figure 7 for the modified Walén test. The test results and other Alfvénic parameters are shown in the same format used in Figure 7  coefficient (from 0.554 to 0.856), compared with Method 3-a. At the same time, γ A (from 1.338 to 1.156), σ c (from 0.851 to 0.871), σ R (from 0.145 to 0.072). and E rr (from 0.446 to 0.153), are improved as seen in Figure 7(a) and (d). The Alfvénic parameters of the x, y, z components from the two methods are listed in Table 6. γ c and of all three components are significantly improved, especially for DV x versus DV x A . The Alfvénic parameters obtained by Method E2-b are closer to 1 (or 0) than those obtained by Method 3 and E2-a.
The results of the modified Walén test of Case 2, Case 3, and Case 4 are shown in Figures 8-10 and Table 7-9 with same format as Figure 7 and Table 6, respectively. For Case 2, γ c (from 0.556 to 0.936) and the Walén slope (from 0.798 to 0.890) are closer to 1, as obtained by Method E2-b, relative to Method 3-a. There are few changes in γ A (from 0.843 to 0.857), σ c (from 0.972 to 0.979), and σ R (from 0.085 to 0.077), and obvious optimization in E rr (from 0.382 to 0.082), as seen in Figures 8 (a) and (b). Compared with Method 3-b, Method E2-b achieved a higher correlation coefficient, with values of 0.936, 0.887, 0.947, and 0.954 for the velocity vector and x, y, z components, respectively, which are also higher than that of method E2-a.  Table 7. Moreover, Method E2-b has a minimum value of both σ R and E rr among the results of modified Walén tests.
For Case 3, Method E2-a produces a Walén slope with a value of 0.960, which is close to 1, and Method E2-b produces an optimal γ c =0.891 [0.902, 0.883, 0.889], .020 [1.090, 1.007, 0.999] for the velocity vector and components, respectively. However, in terms of Figure 9, neither Method 3-b nor Method E2 improve the values of γ A , σ c , σ R , except for E rr (from 0.292 to 0.068). The kinetic energy density is less than the magnetic energy density with a ratio of 0.530 ∼0.626, and the normalized residual energy is greater than 0.250.
For Case 4, the slopes are high, with values of 0.966, but γ c are not so high, with values of 0.734 of unfiltered DV versus DV A , as seen in Figure 10 Table 9. The values of g A , σ c , σ R have few changes from Method 3-a, from Method 3-b to Method E2-a, and from Method E2-b, while E rr is improved from 0.253 to 0.031 by Method E2-b.

Discussions
Four cases of Alfvénic fluctuations in high-speed solar wind are studied in this work. There are strong correlations between the observed plasma velocity and magnetic fluctuations (γ c > 0.9) in all four cases, but their amplitudes match to different degrees. The slopes of their least-squares regression are not stationary, which represents the ratio of ( ) -V V HT versus V A . In order to obtain better and more stationary slopes, two   We found that the Walén relation is better satisfied by fluctuations for the trailing portion of the stream (Case 2 and Case 4) than the leading stream (Case 1 and Case 3), although the correlations between fluctuations of magnetic field and velocity are very close to 1 in each case. It is generally considered that this is possibly due to the influence of strong compression at the leading of high-speed streams. However, the Walén slopes and ratio s s obtained by method E1 are significantly closer to 1 than those for Method 1 and Method 2 in Case 1 and Case 3. This indicates the importance of evaluating the proper HT frame for the Walén test. Additionally, the regression slopes of the x, y, z components are quite different for all three methods, which may be a potential problem for deeper and more comprehensive research on Alfvénic fluctuations with discontinuous structures.
On the other hand, the time difference data of V and V A are analyzed in the modified Walén test by the CHYL methods (Method 3-a and . The only difference between these two methods is that the former uses the original data while the latter uses the five-point smoothed data. The slopes from the original data are almost close to 1 (g >  greater than 1, and the correlation coefficients of DV i and DV i A are relatively low, with values of about 0.5∼0.7. If smoothing data improve these Alfvénic parameters, this likely is due to high-frequency components irrelevant to Alfvénicity. Specifically, the amplitudes of the high-frequency components are reduced and the results from Method 3-b are obviously closer to 1 than those of Method 3-a, which uses smoothed data. There is some artificiality in choosing the filter band for smoothing. In Section 3.3, the EEMD methods (Method E2-a and Method E2-b) are applied for the modified Walén test using different IMF combinations as the reconstructed Alfvénic fluctuations. Compared with the CHYL methods, the IMF group of C 2 ∼R presents results almost similar to those from the five-point smoothed data, but the IMF group of C 3 ∼C N produces the optimal Alfvénic parameters substantially, especially the correlation coefficients and the ratios of deviations. This means that EEMD can efficiently eliminate high-frequency components and vary the background from the original observation, which is irrelevant for Alfvénic fluctuation. As a result, the time differences of the IMF groups from EEMD are more likely suitable for the modified Walén test, even for Case 3 with a more complex HT frame. These comparisons demonstrate that Method E2-b can be used for the modified Walén test as effectively as Method 3, by virtue of the significant affect on noise reduction and superior Alfvénic parameters.
We would like to point out that more fluctuations need to be considered to analyze the robustness of EEMD methods, especially for complex scenarios such as spikes or largeamplitude discontinuous structures existing in the selected period.

Conclusions
AFs are common physical phenomena in solar wind streams, with a distinctive feature in which the fluctuations of plasma velocity and magnetic field are parallel and proportional to each other in observations. But there are some uncertainties in identifications of Alfvén waves based on single-satellite observations at 1 au. This is not only due to the contamination of other propagating waves and non-propagating magnetic structures, but also because of errors caused by data processing.
The Walén relation, as a criterion for identifying Alfvén waves, is usually tested in a constant HT frame where the local solar wind plasma is coherent with an interplanetary magnetic field in a steady medium. But a constant HT frame may be invalid in some complex cases, such as when there is a timedependent or multiple HT frame (Gosling et al. 2009). Considering this, Alfvén waves also can be identified through a modified Walén test using time difference data in the CHYL method (Chao et al. 2014), regardless of the HT frame and other uncertainties. This method can be used to identify the Alfvén waves better, using smoothed data.
In this article, we propose a new Walén test based on selfadapting data analysis technology, the EEMD method, which is intended to analyze the nonlinear and nonharmonic data. In the EEMD process, an (ensemble) average decomposition of signals adds finite-amplitude white noise that is treated as IMFs, and the original data are decomposed into a series of IMFs and a monotonous residual. For comparison, the EEMD process is performed in two ways: the first way is a conventional Walén test based on the HT frame (Method E1); the other is a modified Walén test independent of the HT frame (Method E2).
Parameters related to Alfvénicity are calculated for comparison between different methods, including the Walén slope, correlation coefficient γ c , ratio of the deviations , and the Alfvén ratio γ A , the normalized cross helicity σ c , the normalized residual energy σ R , and the average of the SD ratio E rr . Except for σ R and E rr , which need to be as close to 0 as possible, the other parameters mentioned above imply purely Alfvén waves if they are very close to 1. Four cases of largeamplitude Alfvénic fluctuations are analyzed based on singlesatellite (WIND) observations in a high-speed solar wind stream.
For the conventional Walén test, we extract a time-dependent HT frame from the decomposed residual of the plasma and Alfvén velocity deviation (V i −V Ai ) via EEMD. It is found that Method E1 has superior Alfvénicity. Most of the parameters are optimized, closer to 1 than is seen for the methods with a constant HT frame, while σ R and E rr are closer to 0 as expected.
In the modified Walén tests, it is found that the highfrequency components and slow variation of the background that are unrelated to Alfvénicity can be naturally eliminated from the original data via EEMD. Using the reconstructed fluctuations of the plasma velocity and Alfvén velocity (combinations of IMFs), Method E2 obtains results similar to those of the CHYL method using the smoothed data. And method E2 is more flexible due to its use of the selected IMFs obtained from EEMD. Most of the Alfvénic parameters for the four studied cases are optimized, except for σ c and σ R , which occasionally get a little bit worse.
In summary, according to the results for the four cases studied in this article, we provide an new and alternative approach to the Walén test for identifying Alfvén waves in solar wind: the selfadapting EEMD technique. The EEMD method not only evaluates a time-dependent HT frame velocity, but also produces a higher degree of Alfvénic parameters in either the conventional or modified Walén test. There will be more applications for EEMD for Alfvénic fluctuation observations in the future, due to it its ability to decompose and reconstruct nonstationary signals in the time-frequency domain. Furthermore, the reconstructed fluctuations from EEMD have detailed local properties in narrow frequency bands, which have rarely been discussed before, making it possible to better understand the physical essence of Alfvénic fluctuations in the solar wind.