The occurrence rate of earthward-propagating dipolarization fronts (DFs) is investigated in this paper based on the 9 years (2001–2009) of Cluster 1 data. For the first time, we select the DF events by fitting the characteristic increase inBzusing a hyperbolic tangent function. 303 earthward-propagating DFs are found; they have on average a duration of 4 s and aBz increase of 8 nT. DFs have the maximum occurrence at ZGSM ≈ 0 and r ≈ 15 RE with one event occurring every 3.9 hours, where r is the distance to the center of the Earth in the XYGSM plane. The maximum occurrence rate at ZGSM ≈ 0 can be explained by the steep and large increase of Bz near the central current sheet, which is consistent with previous simulations. Along the r direction, the occurrence rate increases gradually from r ≈ 20 to r ≈ 15 RE but decreases rapidly from r ≈ 15 to r ≈ 10 RE. This may be due to the increasing pileup of the magnetic flux from r ≈ 20 to r ≈ 15 RE and the strong background magnetic field at r <∼13 RE, where the magnetic field changes from the tail-like to dipolar shape. The maximum occurrence rate of DFs (one event per 3.9 hours) is comparable to that of substorms, indicating a relation between the two.
 A recent superposed epoch analysis of DFs, which is based on the THEMIS measurements of six events, indicates that at the DF there is typically a decrease of 50% in the plasma density and ion pressure, a increase of 200%–300% in the high-energy electron flux and electron temperature, and an enhancement of ∼5 mV/m in the duskward and earthward electric field [Runov et al., 2011]. The scale of these gradients is comparable to the ion thermal gyroradius. Schmid et al. reviewed 7-year (2001–2007) Cluster data and found 107 DF events in the magnetotail, among which 69 events have positive minimumBz values at the foot of the DF boundary and 38 events have negative minimum Bz values. They concluded that the thickness of the DF is on average 1.8 ion inertial length (c/ωpi), and the duration of the DF tends to be decreasing with increasing flow velocity. The statistical relation between the DF and the bursty bulk flow was examined using the Geotail [Ohtani et al., 2004] and Cluster [Takada et al., 2006] data. Although these statistical studies (or superposed epoch analyses) investigate many properties of the DF, the occurrence rate of the DF is still not revealed. Also the criteria for distinguishing the DF in the previous studies [e.g., Schmid et al., 2011] are rather general and may include some other structures.
 In this paper, we select the DF events by fitting the characteristic increase of Bzwith a hyperbolic tangent function. We concentrate on the occurrence rate of the earthward-propagating DFs as it can help us to understand the electron acceleration behind the DF.
2. Selection Criteria of DF
 9 years (2001–2009) of Cluster 1 data, particularly the CIS ion velocity data [Rème et al., 2001] with resolution of 4 s and the FGM magnetic field data [Balogh et al., 2001] with resolution of 0.2 s, are used in this study. As Cluster rarely reaches the magnetotail after 2010, the period from 2001 to 2009 almost entirely covers the Cluster observations in the tail. The data from Cluster 2–4 are not considered as they may lead to repeated counts of DF events in the statistics. For the typical flow velocity (250 km/s) and ion inertial length (500 km, calculated by assuming ni ≈ 0.2 cm−3) in the plasma sheet, the DF with a scale comparable to c/ωpihas a duration of ∼2 s, so that the 0.2-sec resolution data of the magnetic field used in this study is sufficient to display the main features of the DF. We describe the parameters in the Geomagnetic solar magnetospheric (GSM) coordinates throughout the paper unless stated otherwise.
 Similar as previous studies [e.g., Schmid et al., 2011], we consider only the DFs in the region −10 > XGSM > −20 RE, −12 < YGSM < 12 RE and −5 < ZGSM < 5 RE. In this region, the earthward propagation of the DF can be distinguished using the flow velocity, as the DF is a tangential discontinuity [Fu et al., 2012]. To detect the DF events, the data of magnetic field and ion flow velocity are divided into windows of 3 min and the window is slid with a step of 1.5 min. The detection of the DF is done in these windows. We first use the criteria given by Sigsbee et al.  and Schmid et al.  to identify the dipolarization events, and then select the DFs from these events. The DF is a special type of dipolarization events as the former (DF) has a typical duration of several seconds while the latter (dipolarization event) can last from near zero to more than 10 min [Sigsbee et al., 2005]. Both DFs and dipolarization events are characterized by the significant increase of BZ. The criteria for identifying the dipolarization events include a large plasma beta (max(β) ≥ 0.5), large flow velocity (max(Vi) ≥ 150 km/s), large inclination angle ( ) of magnetic field (max(θ) ≥ 45°), large change of Bz (max(Bz)−min(Bz) ≥ 4 nT), and large change of inclination angle from minimum Bz to maximum Bz (θ2–θ1 ≥ 10°) in each window, as listed in the right column of Table 1, where the criteria given by Schmid et al.  are also presented for comparison. In each window, the time of the “potential DF” is determined as . Since the window has the width of 3 min and is slid by 1.5 min, data at every time instance have been checked twice, and some dipolarization events might be counted twice. To avoid this “double-counting”, we removed one of the events if the two are separated by less than 30 s.
Table 1. Selection Criteria of DFs in this Study and a Comparison With the Previous Study
 To pick up the steep DFs from the events obtained using the criteria mentioned above, we fit the magnetic field Bz in each window using the hyperbolic tangent function
where Δt = t − tDF represents the interval from 60 s before to 15 s after the DF. In each window, the fitting gives coefficients a, b, c, and a standard error σ. They denote, respectively, the increase of Bz at the DF (a), the duration of the DF (b), the magnitude of the background Bzduring the 1-min interval before the DF (c), and the standard error of the fitting (σ). The duration of the fitted DF (b) covers 80% of the increase of Bfit. For a typical DF event, the fitting gives a large value of a and a small value of b and σ. By requiring the fitting coefficients a > 4 nT, b < 8 s, and the standard error σ< 2.5 nT, we obtained 303 earthward-propagating DFs from those detected dipolarization events.
Figure 1 shows three DF events measured by Cluster 1 on August 15 2001 (Figure 1a), October 2 2006 (Figure 1b), and October 4 2006 (Figure 1c), together with the hyperbolic tangent fit (blue lines). The fitting coefficients are also displayed in each panel. As the magnetic field is typically steady before the DF while disturbed after the DF, we fit the magnetic field from 1 min before to 15 s after the DF rather than during the whole period (1 min before to 1 min after the DF). The DF observed on August 15 2001 is steep so the fitting gives a small value of b = 1.14 s. On October 2 2006, the fitting gives a larger value of b = 2.0 s due to the relatively flat boundary. The standard error of fitting on October 4 2006 (σ = 1.25 nT) is much larger than that on August 15 2001 and October 2 2006 as the magnetic field decreases sharply behind the DF (Figure 1c).
3. Statistical Results
Figure 2presents the superposed epoch analysis of the 303 earthward-propagating DF events (black line) during the interval from 1 min beforetDF to 1 min after tDF. The magnetic field data measured by Cluster 1 are shown in Figure 2a. In Figure 2b, these data have been normalized by the magnetic field in the plasma lobe, Blobe, which is calculated from based on the assumption that the total pressure in the plasma sheet (Pthm_sheet + Pmag_sheet) is balanced by the magnetic pressure in the lobe (Pmag_lobe). The red, upper green, and lower green lines in Figure 2 indicate, respectively, the medians, upper quartiles, and lower quartiles of the measured (Figure 2a) and normalized data (Figure 2b). The profile of the medians is similar as that shown by Ohtani et al. [2004, see Figure 5b] except the DF boundary in our study has a small duration (∼4 s). The median magnetic field is stable before but decreases gradually after the DF. At the DF, this median value jumps from ∼4 to ∼12 nT (Figure 2a). We checked the flow velocities at the DFs and found they are on average ∼210 km/s (not shown), meaning that the average thickness of the 303 DFs is ∼840 km, which is about 1.7 ion inertial length, very close to that (1.8 c/ωpi) given by Schmid et al. .
 We plot these 303 DF events in the XYGSM and ZYGSM plane (Figures 3c and 3d) together with the orbit coverage (Figures 3a and 3b) of Cluster 1 during the 9-year tail period to examine the distribution of DF in the magnetotail. The amplitudes of the DFs are indicated by the size of circles. They seems to be largest at −3 >YGSM > 5 RE (see Figure 3c). The time of Cluster staying in the tail (Figures 3a and 3b) is described in units of days. It has a sum of 4695 hours, as mentioned in Table 1. The SC prefers to stay in the southern hemisphere (ZGSM < 0) and tailward region (XGSM < −16 RE) that is consistent with the Cluster apogee, where the spacecraft has a slow orbital velocity.
 The normalization of the DF counts by the time the spacecraft stay in the magnetotail (Figures 3a and 3b), i.e., the occurrence rate, is displayed in Figure 4 in the XY (Figure 4a), ZY (Figure 4b) and Zr (Figure 4c) plane. Here r denotes the distance to the center of the Earth in the XYGSM plane, . Figures 4d and 4e show the median values of the occurrence rates as a function of ZGSM and r. In the XY plane, the occurrence rate of DFs seems to follow the dashed curve; while in the ZY plane, it peaks at ZGSM ≈ 0. This distribution characteristic is much clearer in Figure 4c, where we can see that the largest occurrence rate with value of R ≈ 6.2 events/day is at ZGSM ≈ 0 and r ≈ 15 RE. In Figures 4d and 4e, the median of the occurrence rate is large at ZGSM ≈ 0 and r ≈ 15 RE, but small at r ≈ 10 RE and r ≈ 20 RE.
 Comparing to the previous studies [e.g., Runov et al., 2011; Schmid et al., 2011], we use a larger database (9-year Cluster data and 4695 hours stay in the magnetotail) and the high-resolution (0.2 s) magnetic field data (seeTable 1). The duration of the DF derived from the median profile of the superposed epoch analysis (see red lines in Figure 2) is ∼4 s in this study, much shorter than that given by Schmid et al. [2011, see Figure 3a]. The magnetic field data with resolution of 4 s considered by Schmid et al.  actually cannot capture the details of the DF as there are only one or two data points across the DF boundary. The DF database (303 events) obtained in this study is larger than that (107 events) given by Schmid et al. . One of the reasons is that, to do the timing analysis, Schmid et al.  considered the DFs measured simultaneously by 4 SC; while in this study, we consider only the DFs measured by Cluster 1.
 To our knowledge, this is the first time the DF events are selected based on fitting the characteristic increase in the Z-component of the magnetic field by a hyperbolic tangent function, and it is also the first time that the occurrence rate of the DFs is reported to peak atZGSM ≈ 0 and r ≈ 15 RE. The distribution ZGSM ≈ 0 indicates that the increase of Bz at the DF is most prominent near the central current sheet, consistent with the simulation results of Sitnov et al. [2009, see Figure 14]. From Figure 4e, one can see that the occurrence rate increases gradually from r ≈ 20 to r ≈ 15 RE but decreases rapidly from r ≈ 15 to r ≈ 10 RE. At r ≈ 20 RE, the occurrence rate is about 1.5 events/day, while at r ≈ 10 RE, it is almost zero. This can be explained by the propagation of the DFs. When the reconnection jet moves earthward from r ≈ 20 to r ≈ 15 RE, the magnetic flux is gradually piled up due to the increasing obstruction by the background magnetic field, so that its leading edge, called the DF, is becoming steeper. In the near-Earth region (r < ∼13 RE), the increase of Bz at the DF is not prominent compared to the strong background magnetic field so that many events have been excluded by the criteria θ2–θ1 ≥ 10° and max(Bz)–min(Bz) ≥ 4 nT, as listed in Table 1. The location, r ≈ 13 RE, is approximately the transition region of the magnetic field from the tail-like to dipolar shape. The largest occurrence rate (R ≈ 6.2 events/day) at ZGSM ≈ 0 and r ≈ 15 RE indicates that a DF can be observed every 3.9 hours, comparable to the occurrence rate of the substorm that is one event per 2.75 hours [e.g., Borovsky et al., 1993].
5. Summary and Conclusions
 The 0.2-sec resolution data of magnetic field and the 4-sec resolution data of ion flow velocity, measured by Cluster 1 in the magnetotail during the 9 years from 2001 to 2009, are examined in this study. Based on this, we identify the dipolarization events by requiring a large inclination angle (max(θ) ≥ 45°), large change of Bz (max(Bz)−min(Bz) ≥ 4 nT) and inclination angle (θ2−θ1 ≥ 10°), together with large flow velocity (max(Vi) ≥ 150 km/s) in every 3-min window of investigation. We then use a hyperbolic tangent function to fitBz in each window. By constraining the fitting coefficients, including the jump of Bz at the DF (ΔBz ≥ 4 nT), the duration of DF (tduration < 8 s) and the standard error (σ< 2.5 nT), we obtained 303 DF events during the 9-year tail (−10 >XGSM> −20 RE, −12 < YGSM < 12 RE, −5 < ZGSM < 5 RE) period. The superposed epoch analysis of these 303 DFs indicates that the DF has on average a duration of 4 s and a Bz increase of 8 nT.
 We investigate the occurrence rate of the DFs and find it peaks at ZGSM ≈ 0 and r ≈ 15 RE with one event per 3.9 hours. The maximum occurrence rate at ZGSM ≈ 0 can be explained by the steep and large increase of Bz near the central current sheet, which is consistent with the simulation results of Sitnov et al. . Along the r direction in the XYGSM plane, the occurrence rate increases gradually from r ≈ 20 to r ≈ 15 RE but decreases rapidly from r ≈ 15 to r ≈ 10 RE. This may be associated with the increasing pileup of the magnetic flux from r ≈ 20 to r ≈ 15 RE and the strong background magnetic field at r < ∼13 RE, where the magnetic field changes from the tail-like to dipolar shape. The maximum occurrence rate of the DFs (one event per 3.9 hours) is comparable to that of the substorm (one event per 2.75 hours), indicating a relation between the two.
 To capture most of the “typical DFs”, the coefficients in fitting the magnetic field Bz are set as a > 4 nT, b < 8 s and σ< 2.5 nT in this study. They correspond to the minimum 4-nT amplitude and maximum 8-sec duration of the DF. By relaxing (restricting) this criterion, more (fewer) “DF events” will be found. The DF database (303 events) obtained in this study covers both the solar maximum (2001–2004) and solar minimum (2005–2009) period. It will be a useful tool to understand the magnetotail dynamics including substorm and reconnection.
 We thank M. Volwerk and R. Nakamura for providing the DF list that can be compared with ours. The authors are grateful to Cluster Active Archive for the data used in this study. This research is supported by the Swedish Research Council under grants 2007-4377, 2009-3902 and 2009-4165.
 The Editor thanks Martin Volwerk and an anonymous reviewer for assisting with the evaluation of this paper.