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 The total electron content (TEC) along the raypath is a fundamental control on the quality of computerized ionospheric tomography (CIT) reconstructions. By measuring the frequency shift or phase, a traditional dual-band beacon can retrieve the relative TEC using the differential Doppler technique. However, because of the uncertainty in the phase integral constants, this technique can only approximately retrieve the absolute TEC. In some cases, the TEC errors are very large and significantly degrade the CIT results. A planned Chinese satellite for seismological studies will carry a triband beacon transmitting VHF, UHF, and L band frequencies. Using the differential phase between each pair of frequencies, the phase ambiguity can be resolved and used to determine the phase integral constant, hence greatly improving absolute TEC retrieval. This paper proposes an algorithm for finding the integral constant in the triband beacon technique. The numerical simulations show that electron density images reconstructed from TEC using three frequencies are much better than those reconstructed from TEC using two frequencies. In particular, when the initial electron density used in the iterative scheme is far from the true value, small-scale structures cannot be discriminated using a dual-band beacon but are visible in reconstructions using the three-frequency technique.
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 Since Austen et al.  introduced computerized ionospheric tomography (CIT), it has made great progress [Pryse et al., 1993; Fougere, 1995; Rogers et al., 2001; Kunitsyn and Tereshchenko, 2003; Bernhardt and Siefring, 2006] and has become an important tool for investigating the ionosphere (see reviews by Pryse , Bust and Mitchell , and Cannon ). These experimental achievements have demonstrated that CIT can throw new light on global ionospheric structures and perturbations and can make an important contribution to applications such as communications, navigation, and positioning. A potential new role for CIT is in early warning and monitoring of earthquakes, for which imaging of small-scale ionospheric disturbances and irregularities is of fundamental importance. Unfortunately, the traditional dual-band beacon technique is powerless for this purpose, but techniques based on a triband beacon offer a way forward. The major earthquake in Wenchuan, China, in 2008 lends a sense of urgency to such work, and a particular motivation for this paper is the planned launch of a Chinese satellite for seismology which will carry a triband beacon transmitter.
 Strange ionospheric phenomena associated with earthquakes were first observed prior to the Alaska “Good Friday” earthquake in 1964 [Moore, 1964]. Since then, unusual ionospheric phenomena preceding earthquakes have been observed many times [Pulinets, 1998; Liu et al., 2000; Silina et al., 2001; Pulinets et al., 2003; Pulinets and Boyarchuk, 2004; Parrot et al., 2006; Zhao et al., 2008; Liu et al., 2009], but there are still many questions about the mechanisms causing these effects. Indeed, arguments continue to rage about whether ionospheric disturbances are necessary precursors of earthquakes. The paper by Geller et al.  led to a significant reduction in research into earthquake prediction, but interest was rekindled by Cyranoski's  paper. This change reflects important progress in theory, data, and new techniques; for example, the French microsatellite Detection of Electromagnetic Emissions Transmitted from Earthquake Regions (DEMETER) was launched in 2004 and observed unusual ionospheric variations over seismic regions before the 2004 earthquake on the Kii Peninsula, Japan [Parrot et al., 2006], and the 12 May 2008 earthquake in Wenchuan [Zhang et al., 2009].
 It has been shown statistically that ionospheric perturbations were observed for 73% of earthquakes with magnitude level 5 and for all earthquakes with magnitude level 6 [Pulinets et al., 2003]. On the basis of hundreds of cases observed using ground-based and satellite measurements, Pulinets and Boyarchuk  summarized the major characteristics of ionospheric effects prior to the earthquakes. They showed that the variations in ionospheric foF2 (the critical frequency of the F2 layer) before earthquakes generally have magnitude of ±30%, but they can be more pronounced, even up to 100% in some instances. Typically, they are at middle and small spatial scales. Evidence for ionospheric precursors up to 5 days before an earthquake was found statistically by Liu et al.  after processing continuous GPS total electron content (TEC) data for 20 strong earthquakes (M ≥ 6) in the Taiwan area. On the basis of the observations associated with 35 M ≥ 6.0 earthquakes that occurred in China during the 10 year period from 1 May 1998 to 30 April 2008, Liu et al.  found that TEC above the epicenter often had pronounced variations 3–5 days before 17 M ≥ 6.3 earthquakes. On 9 May 2008, 3 days prior to the Wenchuan earthquake (M = 8.0), ionosondes recorded an unusually large enhancement of the maximum ionospheric electron density at the F2 peak (NmF2) during the afternoon to sunset sector over Wuhan and Xiamen [Zhao et al., 2008] and Chongqing, China, which are close to the earthquake epicenter. The average increase at these stations was about 2–3 times that expected on a geomagnetically quiet day (the Kp was less than 2) relative to the median May value. GPS TEC and electron density profiles measured by the six microsatellites making up the FORMOSAT-3/COSMIC constellation also observed ionospheric anomalies during the 2008 Wenchuan earthquake [Liu et al., 2009].
 At present, seismoionospheric data mainly come from the rather sparsely distributed network of ground-based ionosondes since topside profiles are not available after the demise of satellites carrying topside sounders, such as Alouette 1 and Intercosmos-19. As a result, only a few complete ionospheric vertical profiles are available, even when topside sounding data are combined with ionosonde data from under the orbit. Since historical observations indicate that many ionospheric parameters, such as foF2, NmF2, F2 layer height (hmF2), sporadic E layer foEs, and TEC [e.g., Pulinets and Boyarchuk, 2004; Zhao et al., 2008; Liu et al., 2009], vary before earthquakes, a high-resolution technique that can give reliable observations of multiple ionospheric parameters is needed. An operational system must also have wide coverage, especially in China, where there are several large and widespread seismic belts. As a result, CIT has once again become the focus of attention, although its shortcomings need to be dealt with. Bernhardt and Siefring  proposed a novel technique using a triband beacon (operating at 150, 400, and 1067 MHz) to remove the problems associated with the traditional dual-band beacon. By using three coherent frequencies, this technique can provide high-accuracy slant TEC to improve the CIT resolution. Although some triband beacon transmitters, such as C/NOFS and COSMIC, have been placed in orbit, further work is needed to improve this technique and especially its use for the complicated issues involved in monitoring ionospheric precursors of earthquakes.
 This paper uses numerical simulations to demonstrate the advantages of three-frequency TEC measurements in the context of CIT imaging. For comparison, two-frequency TEC measurements are simulated using the VHF and UHF band signals of the triband beacon. In section 2, after a brief review of the two- and three-frequency techniques for TEC measurements, an algorithm is proposed that extends the principle of Bernhardt and Siefring  for estimating the phase integral constants. Section 3 deals with numerical simulations carried out for various ionospheric disturbances. Finally, the summary and conclusions are given in section 4.
2. TEC Measurements With Radio Beacons
2.1. Dual-Band Beacon
 TEC can be inferred from measurements of Faraday rotation, differential Doppler phase, and differential group delay. Currently, CIT usually uses TEC retrieved by the differential Doppler technique of measuring the phase or frequency shift, as in this paper. The shifts are introduced by the changes of the radio raypath in the ionosphere, where the phase path is the linear integral of the refractive index. The TEC can be inferred from the information contained in the refractive index of the ionosphere.
 It is assumed that beacon signals are at two coherent frequencies, f1 = n12f0 and f2 = n1n2f0, at base frequency f0, where n1 and n2 are positive integers. Their frequency shifts at a receiver are then
where c is the speed of light in a vacuum and t is time. L1 and L2 are the raypaths in the ionosphere for f1 and f2, respectively, and can be calculated as
where μ is the phase refractive index, ds is the path increment, Ne is the electron density, and r is the location of the receiver. The normalized differential Doppler shift is
 In the receivers, the integration in time will be carried out on the Doppler frequency shift. This integration is used to obtain the differential Doppler phase between two frequencies,
where tL is the time when the receiver locks onto the satellite signal and k1 is a phase integral constant. Since k1 is unknown, PD(t) is a relative value.
 The beacon frequencies are at VHF, UHF, and L band, so the Appleton-Hartree formula under the high-frequency approximation can be used [Davies, 1990]. Generally speaking, the raypath can be regarded as a straight line as long as the ionospheric disturbance is not particularly strong. From (3) and (5), we obtain
Bernhardt and Siefring  proposed the principle of ionospheric sounding by a triband beacon, with an extra carrier frequency, f3 = n22f0, and the interested reader is referred to their paper for the theoretical basis of the technique. After the differential calculation of each pair of frequencies, differential phases are obtained for each of the three pairs:
where ΔP12, ΔP13, and ΔP23 are the absolute values of the differential phases, including the relative phases provided by the receivers and phase integral constants. After some manipulation, one can obtain [Bernhardt and Siefring, 2006]
where Δϕ13 and Δϕ12 are the fractional parts of the differential phases between f1 and f3 and between f1 and f2, respectively, and the mod1 function is used to restrict the values of −Δϕ137 + Δϕ128 to the range 0–1 wavelengths. For the proposed triband beacon, n1 = 3, n2 = 8, and f0 = 16.668 MHz, so equation (8) becomes
where 8.3165 × 1016 is the ambiguity coefficient. Here k2 is referred to as the three-frequency phase integral constant and, like k1, is a positive integer to be determined.
 From (6), we can see that the two-frequency TEC ambiguity coefficient is 1.2995 ×1015, so the three-frequency TEC ambiguity coefficient is about 64 times larger. This is very beneficial for improving the TEC measurement precision since the phase integrals are greatly decreased (k2 ≪ k1) for a much larger ambiguity coefficient and thus are easier to determine. This is the main reason for the superiority of the triband approach. Since Δϕ13 and Δϕ12 can be obtained from the ground receivers, k2 is the only unknown needed to obtain TEC.
 A thorough search of the literature reveals that so far there are no publications on how to find k2 efficiently and correctly. This paper addresses this problem by proposing an algorithm for determining k2 in which, with the help of ground-based receivers, the two-frequency technique is used to give a coarse value of TEC and the three-frequency technique gives the fractional part. Hence, by combining the two- and three-frequency methods, we can retrieve a precise value of TEC. This method is commonly applied in continuous-wave radar for carrier phase ranging using multifrequencies [e.g., Skolnik, 2002]. The range resolution depends on the frequency (or wavelength), with shorter wavelengths giving finer range resolution. However, the full-cycle ambiguity is more serious for shorter wavelengths. To solve this problem, a continuous-wave radar transmits carrier waves at different frequencies. The lower frequencies are used for coarse range measurements, and they are combined with higher frequencies to provide greater precision. The full-cycle ambiguity has also been extensively investigated in GPS carrier phase ranging [El-Rabbany, 2002], and similar approaches are used in many measurement tools, such as the micrometer screw and vernier caliper.
 On this conceptual basis, (6) and (9) for the two- and three-frequency techniques for deriving TEC can be simplified to
where ϕ1 and ϕ2 are the fractional parts of the two- and three-frequency differential phases. Here k1 can be solved using (10) and the multistation method and can then be combined with the fractional part, ϕ1, obtained by using ground receivers to calculate TEC. Since the raypaths are the same, the TEC estimated from (10) can be substituted into (11) to give
 Because the TEC estimated from (10) includes errors, so does the integer value of k2 calculated from (12). However, because the error in the two-frequency measurement by GPS is of the order 1–3 total electron content units (1 TECU = 1.0 × 1016 el m−2) [Wen, 2007], while the ambiguity coefficient of the three-frequency method is 8.3 TECU, the three-frequency method can be used to refine the two-frequency results. In the flowchart of Figure 1, the function floor() is a standard function rounding toward floor, e.g., in MATLAB; that is, it rounds a variable to the nearest integer less than or equal to this variable. As shown in the flowchart in Figure 1, TEC_2 and TEC_3 are variables used to denote the TEC estimates from the two- and three-frequency methods, respectively. The function floor() is used to find the smallest integer for which the two-frequency TEC lies within the three-frequency ambiguity coefficient; this is regarded as a test value and is denoted as k*2. If the error of the two-frequency measurement TEC is less than 4.15 TECU (i.e., half the ambiguity coefficient), then there are three possible values for the three-frequency integer constant: k*2, k*2 + 1, and k*2 − 1. A TEC value can then be calculated by using (9). If the difference between the three-frequency and two-frequency TEC values is less than 4.15 TECU, then the true value of k2 has been found; that is, k2 = k*2. Otherwise, k*2 is overestimated, in which case k2 = k*2 − 1, or k*2 is underestimated, and k2 = k*2 + 1. To validate this method, in section 3 we use the two- and three-frequency methods to solve for TEC and then exploit this value in CIT imaging of small-scale and midscale ionospheric disturbances.
3. Numerical Simulation and Analysis
 Below we use numerical simulations to assess the ability of methods based on a triband beacon to detect seismoionospheric perturbations. A two-dimensional ionospheric profile along 120°E derived from international reference ionosphere (IRI) 2007 is used as the background ionosphere. Two ionospheric disturbances at middle and small scales are embedded in this background between the two humps of the equatorial anomaly. The larger consists of a 30% electron density perturbation of size 400 km, while the smaller one is a 15% perturbation of size 200 km. These parameters are chosen to be consistent with observations before previous earthquakes [Pulinets et al., 2003]. In addition, to account for receiver noise, an error of 0.001 cycles of the phase alignment has been added into the simulations [Wen, 2007; Fridman et al., 2009].
 The simulated model profile with its embedded perturbations is regarded as an ionospheric precursor to an earthquake. From the six stations located along longitude 120°E at intervals of 2° between latitudes 5°N and 15°N, TEC can be calculated using
where TECreal,i is the actual TEC along ray i, aij is the projection length at the grid point j corresponding to the positions of the satellite and receiver, is the electron density at the grid point j according to the simulated ionosphere, and m is the number of grids along the ray i.
 After obtaining the TEC values along the raypaths, the minimum TEC values are found among all rays corresponding to each ground station; these are regarded as the vertical TEC corresponding to each station. The relative TEC can be obtained by subtracting these vertical TEC values from all TEC values calculated at the stations. This relative TEC is regarded as the actual observation, and the phase integral constants can then be calculated for all stations using the multistation method, as shown in Table 1.
Table 1. True Values of Integral Constants and Their Estimates Using the Multistation Method
 Substituting the estimated phase integral constants into (6), the two-frequency TEC measurement can be obtained, and k2 can be calculated by the algorithm illustrated in Figure 1. The three-frequency TEC can then be obtained by substituting k2 into (9). The efficacy of the two- and three-frequency TEC values as inputs into the CIT reconstruction imaging can be compared by using the well-known multiplicative algebraic reconstruction technique (MART) algorithm. This comparison uses the following parameters: the satellite is orbiting at 500 km altitude from 3°S to 23°N latitude; the imaging region covers 0°N–20°N latitude, 120°E longitude; the grid spacing is 10 km (vertical) × 0.5° in latitude (horizontal); and the TEC sampling interval is 0.064°. Reconstruction imaging results are shown in Figure 2, with an initial condition calculated from IRI 2007 at 1200 UT on 12 November 2003 and with a 10% deviation from the true distribution. To aid the comparison, Figure 3 shows the difference between the original and reconstructed distributions. Figures 4 and 5 show similar results for 12 December 2003, with a 25% deviation, and Figures 6 and 7 show similar results for 12 June 2003, with a 42% deviation.
 From Figures 2–7, it can be seen that if the initial conditions deviate greatly from the true values, the reconstruction errors in both the two- and three-frequency techniques increase quickly and the image distortions become more and more severe. However, regardless of how far the initial conditions deviate from the original distribution, the results from the three-frequency technique are superior to those from the two-frequency technique. This is most obvious when the deviation is as large as 42%. In this case, the two-frequency inversion yields significant distortions and artificial perturbations and the small-scale disturbance at the level of 15% can barely be identified, but the small-scale disturbance is still clearly visible in the three-frequency inversion.
4. Summary and Conclusions
 On the basis of the triband beacon principle developed by Bernhardt and Siefring , an effective algorithm for calculating the phase integral constant has been proposed in this paper. It can be used to improve the precision of two-frequency TEC measurements and to refine the two-frequency TEC measurements with errors less than 4.15 TECU, i.e., half of the ambiguity coefficient.
 On the basis of background ionospheres provided by IRI 2007, we simulated 2-D electron density profiles that included disturbances at two different spatial scales. The two- and three-frequency techniques were then used to obtain the absolute TEC where the multistation method was used to deal with the phase integral constant. The approximations embedded in the multistation method led to large errors in the estimation of the absolute TEC when the two-frequency technique was used, but the three-frequency method performed significantly better.
 It was shown that use of TEC values obtained from the three-frequency method as inputs to CIT reconstructions yielded results superior to results obtained by the two-frequency approach, irrespective of how far the assumed initial conditions deviated from the original TEC distribution. The three-frequency technique is especially important for imaging of small-scale structures.
 These findings have particular significance in the context of earthquake prediction since midscale and small-scale ionospheric disturbances are very often observed as precursors to earthquakes, though the mechanisms for their formation are poorly understood. Hence, detection and imaging of such disturbances may be valuable in early warning and forecasting techniques for earthquakes. This is the major motivation for the proposed launch of a Chinese satellite carrying as a key payload a triband beacon for seismological studies. The results in this paper will help to ensure that the measurements to be made by this satellite will yield their full potential.
 This work was supported by the National Natural Science Foundation of China (NSFC) under grants 60771050, 60871076, and 40831062 and the international joint project between NSFC and the Royal Society of the United Kingdom (NSFC-RS) under grant 60811130211.