We present the first calculation of phase and coherence of cross-wavelet transform applied to longitudinally separated L-band equatorial ionospheric scintillation observations received from Geostationary Earth Orbit (GEO) satellites. The phase and coherence analysis were employed on two pairs of observations: (1) São Luís and Rio Branco and (2) Alta Floresta and Huancayo. For these case studies, in statistically significant and high-coherence regions, scintillation observations over São Luís (Alta Floresta) lead that of Rio Branco (Huancayo) by ∼2 to 3 h with a 95%frequency. If L-band scintillation happens over São Luís (Alta Floresta), there is a 95%likelihood that scintillation would happen to the west over Rio Branco (Huancayo) after ∼2 to 3 h, suggesting that a forecast can be made ahead of scintillation occurrences. The phase and coherence relationships between the longitudinally separated scintillation-producing regions can be connected to the large-scale wave structures which are reported to be related to the generation of equatorial spread F and scintillation.
 Comprehending day-to-day variability of equatorial spread F (ESF) and scintillation and forecasting them are challenges of equatorial aeronomy research. Physics-based models and data-driven techniques utilizing ground-based and in situ observations have made contributions toward these ends. An example of a data-driven technique, which is related to the work at hand, developed by Caton and Groves  uses longitudinally separated ground-based scintillation observations. They have investigated the longitudinal correlation of very high frequency and L-band scintillations over the equatorial ionosphere. They have reported strong correlation of scintillation observations for longitudes separated by as much as 30°. Inspired by the work of Caton and Groves , we have here expanded their work by applying a cross-wavelet transform technique on longitudinally separated equatorial ionospheric L-band scintillation measurements.
 In this paper, we present the first calculation of coherence and phase of cross-wavelet transform applied on longitudinally separated pairs of equatorial L-band scintillation observations (Figure 1) thereby to construct tools for predicting scintillation and ESF plasma irregularities westward of L-band signal receiving sites. Application of coherence and phase calculations have been used in climate research to forecast El Niño–Southern Oscillations (ENSO) and Indian summer monsoon [Torrence and Webster1999] and also for investigating the behavior of the Madden-Julian-Oscillation with respect to ENSO [Whitcher et al.2000] (for example).
 Equatorial plasma irregularities and scintillation forecasts are the main objectives of the Communication/Navigation Outage Forecast System (C/NOFS) satellite mission [de La Beaujardiére et al.2004] and the Scintillation Network Decisions Aid (SCINDA) network [Groves et al.1997; Caton and Groves, 2006]. We believe that the results of this research will contribute toward attaining the objectives of the C/NOFS satellite and SCINDA.
2 Data Sources and Analysis Techniques
2.1 The Scintillation Index
 Equatorial spread F, which occurs mostly at night, comprises a broadband spectrum of field-aligned plasma irregularities which are excited by ionospheric interchange instabilities [Hysell, 2000]. The performance of trans-ionospheric communication and navigation satellites is severely degraded by scintillation due to signal-irregularity encounters [de La Beaujardiére et al.2004; de Paula et al.2003]. The strength of scintillation caused by plasma density irregularities can be measured in terms of the scintillation index . The index is defined as the normalized variance of signal intensity.
 To calculate coherence and phase of a pair of scintillation observations separated by longitude, we use 1 min estimates of L-band signals transmitted from GEO satellites (the correction signals for Wide Area Augmentation System and European Geostationary Navigation Overlay Service [European Commision, 2009]) and observed by the Low-Latitude Ionosphere Sensor Network (LISN) which is described by Valladares and Chau . The synchronized motion of the GEO satellites (PRNs: 120, 135, and 138 are used here) and the ground-based receivers (LISN) facilitate the longitudinal localization of signal propagation path which is important for our analysis.
2.2 Coherence and Phase of Cross-Wavelet Transform
 Defined in terms of the cross-wavelet transform, cross-wavelet coherence function or simply coherence is a parameter used to identify both frequency bands and time intervals within which two signals exhibit common power. The two signals in this case are L-band scintillation observations over two locations in South America separated by longitude (Figure 1). Coherence represents the covariance between the two signals as a 2-D time-frequency image. Mathematically, the coherence function can be described in terms of the cross-wavelet transform as [Sello and Bellazzini, 2000]
where , the cross-wavelet transform , ∗ denotes complex conjugation, and φi and φk stand for distinct longitudes i and k. The parameters and are wavelet scale and translation factors, respectively. stands for a wavelet transform in Fourier space at longitude φi [Shume et al.2013],
 The above integral represents the inverse Fourier transform of , where is the Fourier transform of the scintillation signal as a function of time . Here we employed the Morlet wavelet, where , ω∘=6.
 The phase or the relative phase difference , in this specific case, produces a local measure of the delay between the two longitudinally separated scintillation observations in time-frequency image highlighting a coherent behavior. The phase information (lead/lag) can be derived from the imaginary and real parts of [Torrence and Webster, 1999]
where . indicates that longitude φi leads longitude φk.
 This study makes use of coherence and phase of longitudinally separated L-band scintillation observation over the equatorial ionosphere thereby to provide information that could be used to develop a scintillation forecast system westward based on observations in the east. Statistically significant regions of the cross-wavelet spectra (compared to a background red-noise spectrum) will be employed for our analysis and interpretations. This technique simultaneously determines the phase and coherence of scintillation-producing regions separated by longitude. As a result, the analysis provides phase (lead/lag) information of highly coherent structures with a measure of statistical significance. Computing the phase and coherence of longitudinally separated scintillation-producing regions could be important in light of the longitudinal connection of the regions by the large-scale wave structures.
2.3 IPPs of GEO Satellites Used in the Study
 Figure 1 shows the geographic coordinates of the L-band signal receivers used here. Two longitudinally separated scintillation observations are required for the phase and coherence analysis. We used the pairs (1) São Luís (C1) and Rio Branco (C3), whose longitudes are separated by ∼23°, and (2) Alta Floresta (C2) and Huancayo (C4), longitudinal separation ∼21°. The locations of the receivers are close to the magnetic equator (red curve, Figure 1). The longitudinal separations of the pairs are better represented by longitudinal difference of the IPPs of the respective PRN transmissions. The coordinates of the IPPs of GEO satellite signals (PRNs 120, 135, and 138) used in the study at 300 km altitude are shown in Figure 1. Our analysis mostly employs the following PRN transmission pairs:
 PRN 138 (São Luís) and PRN 135 (Rio Branco), whose IPP separation is ∼22.6°
 PRN 138 (São Luís) and PRN 138 (Rio Branco), whose IPP separation is ∼21.3°
 PRN 120 (Alta Floresta) and PRN 138 (Huancayo), whose IPP separation is ∼23.4°
 PRN 138 (Alta Floresta) and PRN 135 (Huancayo), whose IPP separation is ∼21.9°
 The longitudinal separations of the IPPs do not vary with time (for GEO sat) hence GEO satellite signals are suitable for our analysis. The longitudinal separation of the IPPs are very close to the geographic longitudinal separation of the receivers (Figure 1).
3 Results and Interpretation
 Availability of simultaneous L-band scintillation data and the receiving station's vicinity to the magnetic equator allow the choice of the following pairs of stations for our case studies: (1) São Luís and Rio Branco (September to November 2012; March to April 2012) and (2) Alta Floresta and Huancayo (September to November 2011).
3.1 São Luís and Rio Branco
 Figures 2a and 2b present the signal as a function of solar local time on 13/14 October 2012 for São Luís and Rio Branco, respectively. These two signals are from transmissions PRN 138 (São Luís) and PRN 135 (Rio Branco). The corresponding power spectra are plotted in Figures 2c and 2d showing a common power around 130 min. The coherence and phase of the two signals are shown in Figures 2e and 2f.
 Let us focus on the contour curves which bound the statistically significant regions in Figures 2e and 2f. The statistically significant regions are determined by comparing the cross-wavelet spectral power (cws) to a red-noise spectrum (rs), that is, using the ratio . In those regions the coherence is high, mostly close to 1.0 (Figure 2e). The contour curves enclose regions where the cross-wavelet power spectrum is 2, 4, and 6 times larger than a red-noise spectrum, going from the outside contour to the inside. The cones of influence (where edge effect becomes important) in Figures 2e and 2f (shown by black solid lines) are far away from the region of interest. The phase difference is shown in Figure 2f. In regions bounded by the contour curves and corresponding to high coherence, the phase varies very gently from about +π/4 to about +π/6, showing a lead of scintillation activities for observations in the east (São Luís) compared to the west (Rio Branco).
 The phase and coherence calculation has been repeatedly applied on 20 days of L-band scintillation observations from the same pair of stations, São Luís and Rio Branco. The 20 days were selected from the dates, 19 September 2012 to 6 November 2012, where simultaneous scintillation measurements (without dropouts) were available from the LISN database. Horizontal cuts (10 min wide) containing the bands of frequencies through the statistically significant regions of the coherence and phase graphs (figures similar to Figures 2e and 2f were generated) were made for the 20 selected days. All the horizontal cuts of the phases and coherences for the 20 days are plotted in Figure 3a. The left y axis shows the phase (red curve), and the right y axis shows the coherence (green curve). The blue dotted vertical lines in this figure bound the statistically significant regions.
 In Figure 3a, in regions where the coherence is greater than 0.7, the lead is mostly about 1.0 to 3.0 h. The phase is converted to lead/lag based on the fact that the phase variations of ±π correspond to ±12 h. Figure 3c shows the frequency of occurrence of lead/lag in a histogram. The histogram is produced using only a section of the lead/lag data bounded by the blue vertical lines in Figure 3a. The lead/lag ranges from about −1.15 to 3.0 h. The most frequent lead times, however, are from about 2.0 to 3.0 h.
 These results may have implications for scintillation forecasting. The phase and coherence calculations presented in this paper could provide inputs to a system for forecasting scintillation westward based scintillation observations in the east. The histogram in Figure 3c shows that in ∼95%of the cases, the L-band scintillations observed over São Luís are ahead by about 2 to 3 h to that of scintillations observed over Rio Branco. São Luís is to the east of Rio Branco (Figure 1). The implication here is that if L-band scintillation happens over São Luís, the likelihood that scintillation will occur in about 2 to 3 h over Rio Branco is ∼95%. That is, a forecast can be made about 2 h ahead of scintillation occurrences to the west. Actually, the occurrence likelihood gets well above 95%, if lead times include the time range from about to 3 h (Figure 3c). Local sunset occurs about 1 h later in Rio Branco relative to São Luís. The scintillation occurrence delay peaks at 2 to 3 h, implying different local time scintillation onsets to the west. The various components of the geophysical system work in concert to generate L-band scintillation that occurs with high likelihood in the west 2 to 3 h after scintillation occurs in the east.
 Figure 4 shows phase and coherence analysis of L-band scintillation (18 days of simultaneous data) over São Luís and Rio Branco for a transition season (1 March to 15 April 2012 [Aarons, 1993]). Figure 4b shows that in statistically significant regions, the dominant lead is around the 2 to 3 h range.
3.2 Alta Floresta and Huancayo
 To test the validity of our analysis, we applied the technique over another equatorial pair of stations in South America, namely, Alta Floresta and Huancayo (Figure 1) which have comparable longitudinal separation as the pair São Luís and Rio Branco. Like the case study (São Luís and Rio Branco), the coherence and phase analysis were applied on 20 days of scintillation data measured over the Alta Floresta and Huancayo pair. The 20 days were selected from the dates, 30 September 2011 to 12 November 2011, where simultaneous measurements were available from the LISN database. Figure 3b shows the resulting lead/lag and coherence graphs as a function of time. In regions of coherence above ∼0.7, the lead is mostly concentrated around 1.0 to 3.0 h.
 The histogram in Figure 3d clearly shows that the most frequent leads are concentrated from about 2.0 to 3.0 h. Around the left edge, the histogram also shows a lag of about −3.5 to −2.0 h. Those lags actually correspond to the low coherence region (≲ 0.2) in Figure 3b (green curves). The physical implication is that if L-band scintillation happens over Alta Floresta, the likelihood that scintillation will occur in about 2.0 to 3.0 h in Huancayo (to the west) is about 95%. A forecast of about 2.0 to 3.0 h can be made for L-band scintillation monitors ∼20° to the west of Alta Floresta.
 A natural question which comes out of the above analyses is: How are the calculated strong coherence and the calculated phase differences (lead/lag) related to longitudinal correlation of the generation mechanism of plasma irregularities? In recent years, a great deal of effort has been made to understand the relationship between the large-scale wave structures (LSWS) and the development of equatorial spread F and scintillation. The LSWSs are plasma density perturbations in the bottom-side F region; they are aligned to the geomagnetic field and their zonal wavelength can be several hundreds of kilometers; the LSWSs have been reported to have a closer connection and play crucial role in the generation and occurrences of equatorial spread F than the post-sunset layer rise of the F region (for example, [Thampi et al.2009; Tsunoda et al.2011]). The LSWSs reported in these works cover a wide longitude region (about 30°). The phase and coherence relationships between longitudinal separated scintillation-producing regions discussed in this report could be connected to the LSWSs which are prevalent in the equatorial ionosphere and which are reported to be related to generation of equatorial spread F and scintillations.
 We investigated the dependence of the magnitude of coherence on the strength of scintillation. For this purpose, we categorize the scintillation index used in our study into and (these definitions, respectively, correspond to the INACTIVE (quiet) and ACTIVE scintillation nights by Caton and Groves ). Note that the background scintillation values are generally between 0.05 and 0.1. Our cross-wavelet analysis shows that the value of the coherence over the two pair of stations were above 0.7 in 95% of the cases. The rare low coherence values do not show preferential level of scintillation. The analysis also indicates that the coherence is independent of scintillation levels.
5 Summary and Conclusions
 This report presents the first calculation of phase and coherence of cross-wavelet transform applied on two pairs of longitudinally separated L-band scintillation observations received from GEO satellites during September to November and March to April over South America: (1) São Luís and Rio Branco and (2) Alta Floresta and Huancayo. The phase and coherence analysis allows us to build scintillation prediction system westward of observation sites. In statistically significant regions with high cross-spectral coherence, the most frequent distributions show that scintillation observations over São Luís (Alta Floresta) leads that of Rio Branco (Huancayo) by about 2.0 to 3.0 h. In other words, if L-band scintillation occurs in the east over São Luís (or Alta Floresta), the likelihood that scintillation will be observed to the west over Rio Branco (or Huancayo) after about 2.0 to 3.0 h can be about 95%. The phase and coherence relationships between longitudinally separated ionospheric scintillation presented here could be connected to the LSWS which are reported to be related to generation of equatorial spread Fand scintillation.
 The research was carried out at JPL, California Institute of Technology, under a contract with NASA. E. Shume thanks the NPP administered by ORAU under a contract with NASA. We thank Dr. C. E. Valladares for providing the LISN data.
 The Editor thanks Frederick Rich and an anonymous reviewer for their assistance in evaluating this paper.