Ionospheric scintillation can affect not only satellite communications but also satellite measurements of the ionosphere, such as the measurements of total electron content (TEC) made using Transit-like signal transmissions. In this work a numerical phase-screen propagation simulation is applied to study the accuracy of the measurement of TEC made with Transit-like signals. To measure TEC, the phase from coherently related 150 and 400 MHz tones is combined to obtain an estimate of the phase imparted to a propagating signal by transmission through the ionosphere. This phase is processed to remove ambiguities of 2π and converted to a relative value of TEC. However, scattering caused by ionospheric irregularities produces amplitude fading and phase variations that degrade the TEC measurement in several ways. Rapid phase variations that may be associated with deep fades can degrade the phase-ambiguity removal process. Fresnel filtering causes scintillation when intrinsic ionospheric variations are of the size of the Fresnel zone and smaller. Receiver noise also acts to degrade the phase measurements and is especially important during the deepest fades. This paper considers the effects of all these processes and develops rule-of-thumb qualifiers to assure the accuracy of Transit-like measurements of TEC.
 Ionospheric tomography [Austen et al., 1988] has been applied during the last several decades to image the in situ structure of large-scale ionization. Measurements of the total electron content (TEC) along lines of sight from a satellite to multiple ground-based receivers are generally used as the starting point in tomographic reconstruction [Austen et al., 1988; Mitchell and Spencer, 2002]. Presently, the two principal satellite beacons used for ionospheric tomography are Transit-like satellites and the Global Positioning System (GPS). Transit-like receivers provide the differential phase at transmission frequencies of 150 and 400 MHz, from which the relative TEC is obtained. GPS TEC measurements are obtained from either the time delays or the differential phase between the L1 and L2 frequencies (1575 and 1228 MHz) [Conkright et al., 1997; Breed and Goodwin, 1998; Davies and Hartmann, 1997]. Because the TEC derived from the differential phase is more accurate, it is preferable for ionospheric tomography and is used in most of the reported work.
 Many authors have studied errors in the process of ionospheric tomography [Conkright et al., 1997, and references therein]. Pryse et al.  compare three algorithms that implement different approaches to image reconstruction. Biswas and Na  studied the relationship between optimal image resolution and sky coverage. Andreeva et al.  briefly discuss the effects of refraction and diffraction on tomographic reconstruction and also describe the effects of 2π ambiguities in the basic phase measurements. It is well known that large-scale variations in electron density cause most of the variation in observed phase and therefore TEC [Tereshchenko et al., 2002]. However, small-scale irregularities also cause variations in phase and angular scattering which can impact the measurements.
 In a related form of ionospheric tomography [Tereshchenko et al., 2002; Kunitsyn and Tereshchenko, 1992], data from amplitude scintillation are used to obtain images and the spectrum of small-scale irregularities. The images provide the relative variance of electron density fluctuations. In the work of Tereshchenko et al. , the authors discuss the fact that this reconstruction technique depends on Rytov's method of weak scattering, which is valid only when the value of the relative amplitude fluctuation, σχ2, is less than 0.3.
 Finally, accurate measurement of TEC is needed to improve the performance of single-frequency GPS navigation systems [Afraimovich et al., 2000] that must account for otherwise unknown range delays to compute geographic position. Since VHF, UHF, and GPS L band signals are affected by ionospheric scintillation, it is important to understand the effects of scintillation on the underlying phase measurements used in tomographic processing.
 This paper investigates the accuracy of using Transit-like phase measurements to derive the TEC in the case of ionospheric scintillation when the measured phase includes contributions from small-scale irregularities. Scattering caused by ionospheric irregularities produces amplitude fading and phase variations that degrade the TEC measurement in several ways. Rapid phase variations that may be associated with deep fades can degrade the phase-ambiguity removal process. Fresnel filtering causes scintillation when intrinsic ionospheric variations are of the size of the Fresnel zone and smaller. Receiver noise also acts to degrade the phase measurements and is especially important during the deepest fades. This paper considers the effects of all these processes and develops rule-of-thumb qualifiers to assure the accuracy of Transit-like measurements of TEC.
2. Transit TEC Measurement
 Transit-like signals consist of narrow band tones at transmission frequencies of 150 and 400 MHz. To measure TEC, the phase at each of the two transmission frequencies is recorded, modulo 2π. With straight line propagation and no scintillation the phase is the sum of a contribution caused by satellite motion and a contribution due to the ionization that traversed along the propagation path from satellite transmitter to ground-based receiver. The phase contribution from satellite motion is inversely proportional to wavelength, and the contribution from ionization is proportional to wavelength. If the phase is measured simultaneously at both frequencies, unwrapped, and the 2π ambiguities are removed, the two measurements yield two equations in two unknowns; namely the TEC and the range.
In the equations for UHF phase ϕu and VHF phase ϕv, λu and λv are the wavelengths at UHF and VHF, respectively, re is the classical electron radius, (2.82 × 10−15m), NT is the TEC, and R is the range from the satellite to the receiver. With no refraction or scintillation and with sampling sufficient to remove the 2π phase ambiguities, one can measure the change in TEC as the satellite moves in its trajectory above the ionosphere. Only relative TEC is available from this type of measurement unless additional information is available to provide a reference value of TEC at some location.
 Solving for the TEC yields
In some receivers the UHF phase is tracked with a phase-locked loop. In this case the value of ϕu, referred to as the phase reference, can be used in equation (3) to obtain a relationship between the measured value of ϕv and the TEC. In the work here we assume, without loss of generality, that the range R is known perfectly, and we use phase-screen techniques to investigate the accuracy of the calculation of TEC from the simple equation NT = ϕu/λure. Note that many of the results below for the phase reconstruction accuracy are presented as a function of the dimensionless ratio of the spatial sampling interval to the decorrelation distance (or equivalently, the ratio of the temporal sample interval to the decorrelation time); in this case the actual transmission frequency is secondary. Thus the results here are also applicable to GPS L band measurements.
3. MPS Calculation
 The multiple-phase-screen (MPS) simulation technique [Knepp, 1983; Coles et al., 1995] is used to account for RF propagation through the ionosphere. The MPS propagation simulation is quite general and is easily applied to problems involving numerous separated layers of ionization characterized by spatially varying electron density. MPS techniques can handle all levels of ionospheric disturbances from the least severe, where only minor phase fluctuations occur, to the most severe cases of Rayleigh fading, where the scintillation index is unity. A direct solution to the parabolic wave equation is obtained, and the results are exact given the description of the propagation environment. In the calculations reported here the ionization is assumed to consist of long striations, so that a two-dimensional (2-D) model applies. A three-dimensional phase-screen propagation calculation [Martin and Flatte, 1988] would give more accurate results for receivers in the polar region when the geometry could involve propagation along the magnetic field lines. However, the spatial resolution obtained in this work would not be achievable in a three-dimensional calculation because very large amounts of computer resources (time and memory) would be required.
 In the MPS simulation here, 524,288 points are used to represent a grid 6000 km in length. The propagation distance from the ionosphere to the receiver is 300 km. A single frequency is propagated at 400 MHz, corresponding to the UHF Transit tone. The ionosphere is modeled as a one-dimensional phase screen with a power law power spectral density with index 2.7, corresponding to a three-dimensional medium with spectral index of in situ electron density fluctuations of 3.7 (chosen to agree with WBMOD [Secan et al., 1995]). The outer scale is chosen as 1000 km, and the inner scale is 1 m. This nonphysical large value of the outer scale is chosen to permit convenient generation of the phase-screen realizations. The amount of amplitude scintillation depends only on the phase variation at scales on the order of the Fresnel zone size, so large phase variation has no impact on scintillation. The corresponding values of the RMS phase variation that characterize the phase screens as well as variations in electron density are similarly nonphysical because they are based on the outer scale [see Knepp, 1983, equation (11)].
 The use of 524,288 samples over a grid of 6000 km gives a sample spacing of 11.4 m. At a line of sight velocity of 2.5 km/s cutting through the ionospheric layer, this spatial sample spacing corresponds to a temporal spacing of 4.56 ms and a sampling rate of 219 Hz. As a comparison, most existing Transit data receivers for ionospheric tomography use a sampling rate of 50 Hz. To study the effect of reduced sampling rates, the electric field calculated along the MPS grid is decimated. The results of Coles et al.  apply to cases where the MPS grid spacing is varied to determine the accuracy of the MPS calculation itself. In contrast, this work uses high-resolution MPS calculations with many points and investigates the accuracy of the receiver sampling, not the propagation calculation.
 Most of the pictorial examples presented here show only small portions of the received signal calculated along the MPS grid. Statistical quantities (S4, decorrelation distance, phase error, etc.) are calculated over the entire 524,288-point MPS grid and are then averaged over ten independent propagation calculations.
 The MPS code calculates the electric field in the receiver plane after propagation of a 400 MHz CW plane wave 300 km beyond the phase screen. This geometry models the propagation of a signal from a distant satellite through the ionosphere to a ground-based receiver.
 In the results below the statistical properties of the scintillation are related to the errors in the phase reconstruction. The signal propagation simulation is applied to generate densely sampled realizations of received amplitude and phase; the dimension of the simulation (2-D versus 3-D) and match to the actual ionospheric propagation geometry are incidental. However, the realism of the spectrum of the in situ ionization is important since it controls the power in the small-scale irregularities relative to the power in the large-scale irregularities. The small-scale irregularities cause diffraction and scintillation which can disturb the phase measurement.
 If there is no scintillation, the phase of the E-field perfectly matches that of the phase screen. Figure 1 shows an example of the single phase screen that represents the ionosphere (at 400 MHz) as a function of distance along the MPS grid for one quarter of the total propagation grid length. In the figure are two phase screens for comparison: the smooth one for a Gaussian power spectral density (PSD) and the ragged one corresponding to the power law described above. This latter phase screen exhibits small-scale irregularities characteristic of the spectrum used in the rest of this paper.
 Propagation through the screen with the Gaussian PSD produces no scintillation in contrast to an S4 of 0.86 produced by propagation through the power law phase screen. S4 is the normalized standard deviation of the received power given by S42 = 〈(〈I〉 − I)2〉/〈I〉2, where I is the received power and the angle brackets refer to the average over the entire MPS grid. The reconstructed phase of the received E field for the Gaussian PSD exactly matches the phase of the phase screen that represents the ionosphere. In the following, all realizations of amplitude and phase are produced by the power law spectrum of irregularities described above.
Figure 2 shows an example of the amplitude and phase in the case of the power law phase screen. Here the amplitude and phase are shown for a case of moderate scintillation (S4 = 0.62). The E field and phase are shown for the 25-km segment near the beginning of the 6000 km extent of the MPS calculation grid. The upper frame shows the amplitude in decibels; the lower frame shows the measured phase (modulo 2π) and the reconstructed phase, with the phase ambiguities removed. Note the presence of a phase “slip” at a distance of 58 km in the figure. The phase reconstruction process applied in this paper consists of adding or subtracting multiples of 2π so that the resulting phase difference from sample to sample is never more than π.
Figures 3 and 4 show a 1500-km length (1/4 of the MPS grid) and a small 5-km segment at the start of the grid. Each of the figures shows the amplitude (in decibels) and the reconstructed phase of the received electric field and the original ionospheric phase. The ionospheric phase is the phase of the phase screen that represents the ionosphere and is related to the TEC of the ionosphere by ϕ = ∫λreNe(z) dz = λreNT, where Ne(z) is the electron density along the propagation path.
 In Figures 3 and 4 the scattering is weak, S4 = 0.27, and the ionospheric and reconstructed signal phase are similar as expected. As expected, only the small-scale phase variations of the received signal show the effect of scattering. For this level of scattering severity the TEC from Transit measurements would be considered accurate. Note that the use of the word TEC is intended to roughly refer only to the larger-scale size variations in the actual TEC. Small-scale irregularities also produce TEC variations, but the intent here is to heuristically separate large-scale TEC from the small-scale phase scintillation.
 The decorrelation distance is calculated as the distance where the magnitude of the autocorrelation function of the complex E field is down from the peak by a factor of e−1. For each MPS case considered here, 10 realizations are computed and the reported decorrelation distance is the average of 10 individual values. Since 524,288 points comprise each realization, the variance of this quantity is small. For the weak-scattering case shown, the measured decorrelation distance is 794 m, and the ratio of the sample spacing to the decorrelation distance Δx/ℓ0 is 0.014. Consideration of other values of Δx/ℓ0 for this realization is accomplished by skipping grid points. For a moving line of sight through the ionosphere, the spatial variation of the results from the MPS simulation are converted to temporal variations and the ratio Δx/ℓ0 is equivalent to the ratio Δt/τ0, the ratio of the temporal sample spacing to the E field decorrelation time. This feature allows our spatial results to be directly interpreted in terms of temporal quantities of interest in satellite-based ionospheric measurements.
Figures 5 and 6 are similar to Figures 3 and 4 but show two close-ups of the MPS grid representing only a small percentage of the total grid length of the calculated received signal. In this figure the measured value of S4 is 0.62 and decorrelation distance is 271 m, giving a minimum value of Δx/ℓ0 of 0.042. In Figure 5 the deep fading at about 58 km along the grid causes a 2π phase jump relative to the ionospheric phase. This jump occurs because the signal level in the deep fade at that point is small; even the smallest variation in amplitude can then cause rapid phase changes. A comparison of Figures 4 and 6 illustrates that the additional scintillation (which is caused by additional angular scattering) is causing increased disagreement between the ionospheric phase and the reconstructed phase of the received E field.
Figure 7 is in the same format as the preceding figures and shows the amplitude, ionospheric phase, and E field phase as a function of distance along one quarter of the entire MPS grid. In the case shown in the figure, S4 is 1.0, which is characteristic of strong scattering. Note the increased disagreement between the reconstructed phase of the received electric field and the ionospheric phase.
Figure 8 shows the effects on the phase reconstruction and TEC calculation processes of reduced resolution in the sampling of the electric field. In this figure the MPS realization discussed previously for the S4 value of 0.27, representing weak scintillation, is used repeatedly but with different sample spacing. The variations in the sample spacing are obtained by using every point, then every second point, every third point, and so on. The figure shows the TEC in TEC units (1016el/m2) calculated from the reconstructed phase on one quarter of the MPS grid. The ionospheric TEC and the reconstructed TEC is shown for values of Δx/l0 of 0.01, 0.25, 0.75, 1.0, and 2.0. Conventional Transit 50-Hz sampling for this MPS case (with l0 of 794 m and a line of sight velocity of 2.5 km/s through the ionospheric layer) would give a corresponding value of Δx/ℓ0 of 0.63. Thus this level of moderate scintillation would give good TEC measurements using a conventional 50-Hz Transit receiver.
 Note that as the value of Δx/ℓ0 exceeds about 1.0 in the figure, the TEC calculated from the phase of the E field is dramatically different than the TEC of the ionosphere. Certainly at this point the value of Transit TEC measurements would be questionable. For a line of sight velocity of 2.5 km/s the Δx/ℓ0 = 1 curve corresponds to a sampling rate of about 4 Hz. Thus sample rates of 4 Hz or less would be problematic in the measurement of TEC at UHF under conditions of moderate scintillation. Figure 9 gives a close-up of Figure 8 for a 200-km segment at the start of the MPS calculation grid. The curves of TEC versus distance for Δx/ℓ0 of 0.01 and 0.25 overlay the ionospheric TEC; the TEC curve for a value of Δx/ℓ0 of 0.75 is in error by about 1 TEC unit at the end of the 200-km segment. The curves of TEC for Δx/ℓ0 values of 1.0 and 2.0 are significantly different than the ionospheric phase.
Figure 10 has the same format as the previous two figures. Here the case under study represents moderate scintillation where S4 is 0.62. This case was discussed previously in connection with Figures 5 and 6. In comparison to the weak scintillation case illustrated in Figures 8 and 9, the case of moderate scintillation has less accuracy in the phase reconstruction and TEC calculation process.
Figure 11 summarizes the effects of scintillation and sampling on TEC measurement using the Transit phase reconstruction technique. Five curves are shown for S4 values of 0.23, 0.38, 0.62, 0.75, and 1.0. For S4 less than 0.75 and Δx/ℓ0 less than about 0.75, this technique gives good results for the TEC as shown in the figure.
5. Receiver Noise
Figure 12 gives an example of the effects of receiver noise on the phase reconstruction process. The case shown corresponds to the moderate scattering MPS calculation shown in Figure 6 (S4 = 0.62), to which white Gaussian noise has been added to the received signal to give a value of average signal-to-noise ratio of 8 decibels. In the example the combination of a deep fade and noise at a distance of 3.5 km on the plot gives a phase jump of 2π. At this point and beyond the reconstructed signal phase deviates from the ionospheric phase.
Figure 13 shows the same MPS calculation as depicted in Figures 6 and 12, with the exception that the average SNR is 12 dB. For comparison, in Figure 6, there is no noise. From a comparison of Figures 6 and 13 it is evident that the effects of noise on the phase measurements are minimal for an average SNR value of 12 dB. In other calculations of the effects of noise, we have found that there is little degradation to the phase reconstruction process as long as the average SNR is around 15 dB or greater.
 This work utilizes phase-screen simulation techniques to perform a numerical investigation of the accuracy of the measurement processing technique used by Transit-like receivers to compute the TEC due to the natural ionosphere. The results here for the effects of ionospheric scintillation apply in general to receivers that use phase reconstruction techniques based on the measured modulo 2π phase difference between samples. We have found that Transit-like phase reconstruction and the resulting measurements of TEC are reasonably accurate if the scintillation index is less than 0.75, the average signal-to-noise density ratio is greater than about 15 decibels, and the sample spacing is no greater than 0.75 times the decorrelation time of the received signal.
 The encouragement and technical advice of L. J. Nickisch of Mission Research Corporation and Jim Secan of NorthWest Research Associates was invaluable in the preparation of this paper.