Analysis of Plasma Bubble Signatures in Total Electron Content Maps of the Low-Latitude Ionosphere: A Simplified Methodology

Abstract

The ionosphere over the Brazilian region has particular characteristics due to the large geomagnetic declination angle over most of the territory. Furthermore, the equatorial ionization anomaly southern crest is located over the Brazilian territory. In this region, plasma irregularities may arise in the post-sunset hours. These ionospheric irregularities develop in the form of magnetic field-aligned plasma depletions, known as equatorial plasma bubbles, which may seriously affect radio signals that propagate through them. These irregularity structures can cause amplitude and phase scintillation of the propagating signals, thereby compromising the availability, performance, and integrity of satellite-based communication and navigation systems. Additionally, the total electron content (TEC) introduces propagation delays that can contribute to range measurement errors for global positioning system (GPS) users. The ionospheric characteristics change significantly according to the time of day, season, as well as the solar and geomagnetic activities, among other factors. Indeed, the ionosphere is one of the most significant sources of errors in the positioning and navigation systems based on the GPS satellites. Due to these features, there is a strong interest by the scientific community in better understanding and characterizing the ionospheric behavior. In this context, the TEC analysis has wide applicability for space plasma studies and is a well-established tool for investigating the ionospheric behavior and its potential impact on space-based navigation systems. One of the goals of these studies is the generation of TEC maps for a geographic region based on GPS observations. In the present work, some electrodynamic processes of the low-latitude ionosphere are reviewed and the TEC estimation based on GPS measurements is revisited in detail. A methodology aimed at creating the TEC maps is presented and validated by comparison with results from other geophysical instruments, such as all-sky imagers and ionosondes. Finally, examples of the ionospheric behavior displayed by TEC maps during equatorial plasma bubble events and a geomagnetic storm are fully described and discussed.

Appendix: A review of TEC Measurements from GPS Observations

In terms of geodetic positioning, TEC produces an important part of the effects in GNSS signals. In the present section, the relationship between the TEC and the ionospheric refraction for phase and pseudorange measurements will be discussed in more detail. A work that details the methodology of TEC measures from GPS observations that is worth mentioning is Jung and Lee (2012).

Indeed, the basic GPS observables are the pseudorange and carrier phase. A less-used third observable is the Doppler measurement, which represents the difference between the nominal and received frequencies of the signal. Using the difference between the pseudoranges ($P_{1}$ and $P_{2}$) and the carrier phase measurements ($L_{1}$ and $L_{2}$) at two frequencies, it is possible to compute the value of the slant TEC (TEC_s) measured along a straight path from the satellite to the receiver (Hofmann-Wellenhof et al. 2008).

1.1 Ionospheric Refraction and its Relation with TEC

The TEC is the number of electrons in a column with a one-square-meter cross section along a signal path from a GNSS satellite to a receiver antenna (Seeber 2003):

$${\text{TEC}} = \int\limits_{\text{ray}} {N_{\text{e}} } {\text{d}}s\left[ {\frac{\text{el}}{{{\text{m}}^{2} }}} \right]$$

(A.1)

where N_e is the electron density (el/m³). Considering a simplified expression for the index of refraction $n \approx 1 - KN_{\text{e}} /\left( {2f^{2} } \right)$, valid under the assumption of relatively high frequencies, the ionospheric delay (I_f) can be related to TEC through:

$$I_{f} = \frac{K}{{2f^{2} }}{\text{TEC}}$$

(A.2)

where K = 80.62 (m²/s²) represents the ionospheric refraction and $f$ is the carrier frequency. Thus, the magnitude of the systematic error due to ionospheric refraction is directly proportional to TEC and inversely proportional to the square of the carrier wave frequency.

1.2 TEC from Pseudorange Observations

GPS receivers use the C/A and P codes to determine the pseudorange, which measures the optical distance between the satellite and the receiver, including the effects from the medium. The receiver replicates the code generated by the satellite and determines the elapsed time for the propagation of the signal from the satellite to the receiver by correlating the transmitted code and the code replica. The pseudorange is computed by simply multiplying the time offset by the speed of light.

The pseudorange observation equations can be written as (Ciraolo et al. 2007; Carrano and Groves 2009; Seemala and Valladares 2011):

$$P_{1} = \rho + c\left( {\Delta t_{\text{r}} - \Delta t_{\text{s}} } \right) + I_{1} + T + b_{{1{\text{r}}}} + b_{{1{\text{s}}}} + m_{1} + \varepsilon_{1}$$

(A.3)

$$P_{2} = \rho + c\left( {\Delta t_{\text{r}} - \Delta t_{\text{s}} } \right) + I_{2} + T + b_{{2{\text{r}}}} + b_{{2{\text{s}}}} + m_{2} + \varepsilon_{2}$$

(A.4)

where $P_{1}$ and $P_{2}$ are the pseudorange observations corresponding to the frequencies $f_{1}$ and $f_{2}$; $\rho$ is the geometric distance from the GPS receiver’s antenna phase center at the epoch of signal reception to the GPS satellite’s antenna phase center at the epoch of signal transmission; $c$ is the speed of the light; $\Delta t_{\text{r}}$ and $\Delta t_{\text{s}}$ are the receiver and satellite clock errors; $I_{1}$ and $I_{2}$ are the ionospheric delays for frequencies $f_{1}$ and $f_{2}$; $T$ is the tropospheric delay, $b_{{1{\text{r}}}}$ and $b_{{2{\text{r}}}}$ are the instrumental biases for the receiver; $b_{{1{\text{s}}}}$ and $b_{{2{\text{s}}}}$ are the instrumental biases for the satellite; $m_{1}$ and $m_{2}$ are the associated with multipath effects; and $\varepsilon_{1}$ and $\varepsilon_{2}$ are thermal noise components. The indices 1 and 2 represent the frequencies f₁ and f₂, respectively.

Using Eqs. (A.3) and (A.4) to form the difference $P_{2} - P_{1}$, neglecting the multipath and thermal noise terms, and considering that the geometric range, clock error, and tropospheric delay terms cancel, one gets:

$$P_{2} - P_{1} = I_{2} - I_{1} + \left( {b_{{2{\text{r}}}} - b_{1E} } \right) + \left( {b_{{2{\text{s}}}} - b_{{1{\text{s}}}} } \right)$$

(A.5)

$$P_{2} - P_{1} = I_{2} - I_{1} + b_{\text{r}} + b_{\text{s}} .$$

(A.6)

Substituting the ionospheric delay presented in Eq. (A.2) into the pseudorange observation equation, it follows that:

$$P2 - P1 = \frac{K}{2}\left( {\frac{1}{{f_{2}^{2} }} - \frac{1}{{f_{1}^{2} }}} \right){\text{TEC}}_{\text{slp}} + b_{\text{r}} + b_{\text{s}}$$

(A.7)

$${\text{TEC}}_{\text{slp}} = \frac{{2\left( {f_{1} f_{2} } \right)^{2} }}{{K\left( {f_{1}^{2} - f_{2}^{2} } \right)}}\left( {P_{2} - P_{1} } \right) - b_{\text{r}} - b_{\text{s}} \left[ {\frac{\text{el}}{{{\text{m}}^{2} }}} \right]$$

(A.8)

where ${\text{TEC}}_{\text{slp}}$ is the slant TEC calculated using pseudorange measurements.

1.3 TEC from Carrier Phase Measurements

The carrier phase measurements correspond to the phase difference between the received signal (transmitted by the satellite) and the signal generated by the reference oscillator of the receiver. The equations for carrier phase measurements are:

$$\phi_{1} = \rho + c\left( {\Delta t_{\text{r}} - \Delta t_{\text{s}} } \right) - I_{1} + T + b_{{1{\text{r}}}} + b_{{1{\text{s}}}} + \lambda_{1} N_{1} + m_{1} + \varepsilon_{1}$$

(A.9)

$$\phi_{2} = \rho + c\left( {\Delta t_{\text{r}} - \Delta t_{\text{s}} } \right) - I_{2} + T + b_{{2{\text{r}}}} + b_{{2{\text{s}}}} + \lambda_{2} N_{2} + m_{2} + \varepsilon_{2}$$

(A.10)

where $\phi_{1}$ and $\phi_{2}$ are the carrier phase observations corresponding to frequencies $f_{1}$ and $f_{2}$; $\rho$ is the geometric distance from the GPS receiver’s antenna phase center at the epoch of signal reception to the GPS satellite’s antenna phase center at the epoch of signal transmission; $c$ is the speed of light; $\Delta t_{\text{r}}$ and $\Delta t_{\text{s}}$ are the receiver and satellite clock errors; $I_{1}$ and $I_{2}$ are the ionospheric delay for frequencies $f_{1}$ and $f_{2}$; $T$ is the tropospheric delay; $b_{{1{\text{r}}}}$ and $b_{{2{\text{r}}}}$ are the instrumental biases for the receiver; $b_{{1{\text{s}}}}$ and $b_{{2{\text{s}}}}$ are the instrumental biases for the satellite; $\lambda_{1}$ and $\lambda_{2}$ are the wavelength; the integers $N_{1}$ and $N_{2}$ are associated with cycle ambiguities; $m_{1}$ and $m_{2}$ are associated with multipath effects; and $\varepsilon_{1}$ and $\varepsilon_{2}$ are thermal noise components. The indices 1 and 2 represent the frequencies $f_{1}$ and $f_{2}$, respectively.

To use the carrier phase as an observable for positioning, the unknown number of cycles or ambiguity N has to be determined by appropriate methods (Langley 1998). One of these methods will be described below.

Using Eqs. (A.9) and (A.10) to form the difference $\phi_{1} - \phi_{2}$, neglecting the multipath and thermal noise terms, and considering that the geometric range, clock error, and tropospheric delay terms cancel, one gets:

$$\phi_{1} - \phi_{2} = I_{2} - I_{1} + \left( {b_{{1{\text{r}}}} - b_{{2{\text{r}}}} } \right) + \left( {b_{{1{\text{s}}}} - b_{{2{\text{s}}}} } \right) + \left( {\lambda_{1} N_{1} - \lambda_{2} N_{2} } \right)$$

(A.11)

$$\phi_{1} - \phi_{2} = I_{2} - I_{1} + b_{\text{r}} + b_{\text{s}} + \left( {\lambda_{1} N_{1} - \lambda_{2} N_{2} } \right).$$

(A.12)

Substituting the ionospheric delay represented by Eq. (A.2), $\phi_{1} = L_{1} \lambda_{1}$ and $\phi_{2} = L_{2} \lambda_{2}$ into the carrier phase observation Eq. (A.12), it follows that:

$$\phi_{1} - \phi_{2} = \frac{K}{2}\left( {\frac{1}{{f_{2}^{2} }} - \frac{1}{{f_{1}^{2} }}} \right){\text{TEC}}_{\text{sll}} + b_{\text{r}} + b_{\text{s}} + \left( {\lambda_{1} N_{1} - \lambda_{2} N_{2} } \right)$$

(A.13)

$${\text{TEC}}_{\text{sll}} = \frac{{2 \left( {f_{1} f_{2} } \right)^{2} }}{{K\left( {f_{1}^{2} - f_{2}^{2} } \right)}}\left( {L_{1} \lambda_{1} - L_{2} \lambda_{2} } \right) - b_{\text{r}} - b_{\text{s}} - \left( {\lambda_{1} N_{1} - \lambda_{2} N_{2} } \right)\left[ {\frac{\text{el}}{{{\text{m}}^{2} }}} \right]$$

(A.14)

where ${\text{TEC}}_{\text{sll}}$ is the slant TEC calculated using phase measurements, and L₁ and $L_{2}$ are the number of cycles corresponding to the frequencies $f_{1}$ and $f_{2}$.

1.4 Correction of Cycle Slips

Cycle slips are abnormal jumps in carrier phase measurements when the receiver phase-tracking loops experience temporary losses of lock or some other disturbing factor, which must be detected and corrected. Cycle slips can occur due to the failures in the receivers, as well as obstructions of the signal, high signal noise, or low signal strength. The magnitude of a cycle slip may range from a few cycles to millions of cycles (Seeber 2003).

To detect cycle slips, several testing quantities which are based on various combinations of GPS observations have been proposed (Seeber 2003; Hofmann-Wellenhof et al. 2008). Some of these methods depend on single, double, or triple differences of observations. Once a cycle slip has been detected, it can be repaired using a technique that requires an exact estimation of the size of the slip and can be done instantaneously.

Considering a stand-alone GPS receiver, the previous observation equations for carrier phase measurements can be reformulated as (Dai 2012):

$$\lambda_{1} L_{1} = \rho + \lambda_{1} N_{1} - I_{1} + \varepsilon_{L1}$$

(A.15)

$$\lambda_{2} L_{2} = \rho + \lambda_{2} N_{2} - \frac{{\lambda_{2}^{2} }}{{\lambda_{1}^{2} }}I_{1} + \varepsilon_{L2}$$

(A.16)

where $\lambda_{1}$ and $\lambda_{2}$ are the wavelength of the corresponding GPS signal; $L_{1}$ and $L_{2}$ are the received carrier phase observables in units of cycles; $\rho$ is the geometric distance from the GPS receiver’s antenna phase center at the epoch of signal reception to the GPS satellite’s antenna phase center at the epoch of signal transmission; $N_{1}$ and $N_{2}$ are the integer phase ambiguity in units of cycles; $I_{1}$ is the ionospheric delay in units of length, while $\varepsilon_{L1}$ and $\varepsilon_{L2}$ combine the other terms in the previous equations, assumed to be random.

A cycle slip can be obtained by differencing the carrier phase observation equations between two consecutive epochs:

$$\lambda_{1} \Delta L_{1} = \Delta \rho + \lambda_{1} \Delta N_{1} - \Delta I_{1} + \varepsilon_{L1 }$$

(A.17)

$$\lambda_{2} \Delta L_{2} = \Delta \rho + \lambda_{2} \Delta N_{2} - \frac{{\lambda_{2}^{2} }}{{\lambda_{1}^{2} }}\Delta I_{1} + \varepsilon_{L2}$$

(A.18)

where the operator $\Delta$ indicates the differencing between values associated with the current and the last epochs. The known terms in the measurements domain are the carrier phase measurements $L_{1}$ and $L_{2}$. Thus, the cycle slip detection is based on the relation between the cycle slip terms $\Delta N_{1}$ and $\Delta N_{2}$ and the measurement terms $\Delta L_{1}$ and $\Delta L_{2}$. The elimination of the geometric term $\rho$ is a key step for cycle slip detection. The geometry term in Eqs. (A.15) and (A.16) can also be estimated using other measurements unaffected by cycle slips. Concerning the GPS observations, this term can be estimated by pseudorange data:

$$P_{1} = \rho + I_{1} + \varepsilon_{P1}$$

(A.19)

$$P_{2} = \rho + \frac{{\lambda_{2}^{2} }}{{\lambda_{1}^{2} }}I_{1} + \varepsilon_{P2} .$$

(A.20)

The random terms in expressions (A.19) and (A.20) are analogous to $\varepsilon_{L1}$ and $\varepsilon_{L2}$. Differencing the pseudorange observation equations between two consecutive epochs, one finds:

$$\Delta P_{1} = \Delta \rho + \Delta I_{1} + \varepsilon_{P1}$$

(A.21)

$$\Delta P_{2} = \Delta \rho + \frac{{\lambda_{2}^{2} }}{{\lambda_{1}^{2} }}\Delta I_{1} + \varepsilon_{P2} .$$

(A.22)

In comparison with Eqs. (A.17) and (A.18), the differences in the code observation equations lie in the opposite sign of the ionospheric delay term, as well as the much larger thermal noise and multipath errors. Differencing Eqs. (A.17) and (A.21), (A.18) and (A.22), assuming that the ionospheric delay does not substantially change between consecutive epochs (so that their differences can be incorporated into the random components), and rearranging the remaining terms, one gets:

$$\Delta N_{1} = \frac{{\lambda_{1} \Delta L_{1} - \Delta P_{1} }}{{\lambda_{1} }} + \varepsilon_{1}$$

(A.23)

$$\Delta N_{2} = \frac{{\lambda_{2} \Delta L_{2} - \Delta P_{2} }}{{\lambda_{2} }} + \varepsilon_{2} .$$

(A.24)

In expressions (A.23) and (A.24), the terms ε_i (where i = $L_{1}$, $L_{2}$, $P_{1}$, $P_{2}$ and their differences) can be determined from definitions (A.15)–(A.22). Thus, if the first terms of the right sides of Eqs. (A.23) and (A.24) substantially exceed the seconds, there are occurrences of cycle slips, which must be corrected (Blewitt 1990).

Dual-frequency GPS receivers present a twofold superiority over single-frequency receivers for cycle slip detection. First, the geometry term $\rho$ and the non-dispersive errors can be fully eliminated using a geometry-free phase combination. Second, the carrier phase measurements present much lower thermal noise error than the code data (Dai 2012). The geometry-free combination provides:

$$\lambda_{1} \Delta L_{1} - \lambda_{2} \Delta L_{2} = \lambda_{1} \Delta N_{1} - \lambda_{2} \Delta N_{2} - \left( {1 - \frac{{\lambda_{2}^{2} }}{{\lambda_{1}^{2} }}} \right)\Delta I_{1} .$$

(A.25)

The cycle slips occurring either on L₁ or L₂ or both signals can be detected if the following condition holds:

$$|\lambda_{1} \Delta L_{1} - \lambda_{2} \Delta L_{2} | > a \sigma_{\text{comb}}$$

(A.26)

where a is a multiplication factor determining the confidence level,

$$\sigma_{\text{comb}} = \sqrt 2 \sqrt {\lambda_{1}^{2} - \lambda_{2}^{2} } \sigma_{L1} .$$

(A.27)

In the last expression, $\sqrt 2$ reflects the between-epoch differencing and $\sigma_{L1}$ stands for the standard deviation of carrier phase thermal noise.

Dual-frequency measurements also allow a cycle slip determination based on the fact that the removal of the most likely value of cycle slip from the carrier phase data could yield the minimal residuals of geometry-free combination. The formulation of search space of cycle slip candidates is expressed in detail in Teunissen (1995). Removing the cycle slip candidates from the carrier phase data should make the repaired carrier phase data most possibly pass cycle slip detection tests (Dai 2012).

1.5 Leveling of the Carrier Phase with the Pseudorange

The difference (L₂ − L₁) is less noisy than the one provided by (P₂ − P₁), but does not provide the absolute slant TEC. Additionally, the ambiguities in the integer values of L_1,2, known as a cycle slip, often arise. Typically, the cycle slip correction can be made with the aid of the pseudorange difference measurement, which is unambiguous but noisier (Ciraolo et al. 2007; Ma and Maruyama 2003). To retain the accuracy of the slant ${\text{TEC}}_{\text{sl}}$, a baseline B_rs is computed as the average difference between pseudorange-derived ${\text{TEC}}_{{{\text{slp}}_{i} }}$ and phase-derived ${\text{TEC}}_{{{\text{sll}}_{i} }}$,

$$B_{rs} = \frac{{\mathop \sum \nolimits_{i = 1}^{N} \left\langle {{\text{TEC}}_{{{\text{slp}}_{i} }} - {\text{TEC}}_{{{\text{sll}}_{i} }} } \right\rangle_{\text{arc}} \sin^{2} \varepsilon_{i} }}{{\mathop \sum \nolimits_{i = 1}^{N} \sin^{2} \varepsilon_{i} }}$$

(A.28)

with i = 1,2,…, $N$, where N is the number of continuous measurements contained in the arc, and $\varepsilon$ is the elevation angle. The notation $\left\langle {} \right\rangle_{\text{arc}}$ in Eq. (A.28) indicates that the involved values should only be taken over a phase-connected arc (between successive cycle slips) (Ciraolo et al. 2007; Ma and Maruyama 2003). The estimated baseline B_rs is then added to ${\text{TEC}}_{\text{sll}}$ to provide ${\text{TEC}}_{\text{sl}}$ (Mannucci et al. 1998)

$${\text{TEC}}_{\text{sl}} = {\text{TEC}}_{\text{sll}} + B_{rs} .$$

(A.29)

1.6 Differential Code Biases and Absolute sTEC

Once the relative ${\text{TEC}}_{\text{sl}}$ has been determined, subtraction of the satellite and receiver differential instrumental code biases yields the calibrated sTEC:

$${\text{sTEC}} = \left( {{\text{TEC}}_{\text{sl}} - b_{s} - b_{r} } \right)$$

(A.30)

$$B_{i} = b_{\text{s}} + b_{\text{r}}$$

(A.31)

where $b_{\text{s}}$ and $b_{\text{r}}$ are the instrumental differential code biases (DCBs) of the GPS satellite s and receiver r (Kenpankho et al. 2011), respectively, estimated according to the procedures described next.

The differential code bias (B_i) of each receiver–satellite pair is obtained by comparing the hourly averages of uncalibrated TEC values from all satellite and single receiver combinations, using the weighted least mean square fitting (LMSQ) method (Otsuka et al. 2002). That is, the set of parameters $\overline{V}_{k}$ and B_i are estimated by minimizing the squared error E² defined by:

$$E^{2} = \mathop \sum \limits_{i}^{{N_{s} }} \mathop \sum \limits_{k}^{{N_{t} }} W_{k}^{i} \left[ { \frac{{\overline{{({\text{TEC}}_{\text{sl}} )}}_{k} }}{S\left( \varepsilon \right)} - \overline{V}_{k} - \overline{{\frac{1}{S\left( \varepsilon \right)}}} B_{i} } \right]^{2} .$$

(A.32)

In the above equation, k =1, 2, …, $N_{\text{t}}$; i =1, 2, …, $N_{\text{s}}$; $N_{\text{t}}$ is the number of hourly TEC averages; and $N_{s}$ is the number of satellites observed by the receiver. Here, it will be assumed that the hourly average of vertical TEC ($\overline{V}_{k}$) is uniform within an area covered by a receiver. All variables with overlines denote average values. $W_{k}^{i}$ is the weighting function:

$$W_{k}^{i} = \overline{{\frac{1}{S\left( \varepsilon \right)}}}$$

(A.33)

that depends on the slant factor:

$$S\left( \varepsilon \right) = \frac{1}{{\cos \left[ {\arcsin \left( {\frac{{R_{\text{E}} \cos \varepsilon }}{{R_{\text{E}} + H}}} \right)} \right]}}.$$

(A.34)

In the above expression, $\varepsilon$ is the elevation angle and $R_{\text{E}}$ is the Earth radius. It is assumed that the ionosphere is compressed into a thin shell over the peak of the ionospheric F layer. The ionospheric pierce point (IPP) is defined as the intersection of the receiver-to-satellite ray path with the thin shell, which is assumed to have an altitude H between 340 and 400 km (here, $H$ has been kept at 400 km).

Once B_i is calculated using the weighted LMSQ fitting method, it is also possible to estimate the differential code bias of the receiver using Eq. (A.31), if the satellite instrumental code bias b_s is taken from the code bias files ($P_{1} C_{1}$ and $P_{1} P_{2}$) estimated by the Center for Orbit Determination in Europe (CODE).

If the minimum ${\text{sTEC}}_{\text{dailymin}}^{\text{r}}$ (relative TEC after correction of each GPS satellite DCB, but still biased with the receiver DCB) continues to be negative after the application of the LMSQ, it is possible to assume that the receiver DCB is equal to that value (Rideout and Coster 2006; Ciraolo et al. 2007). Generally, the DCB calculated from the ZERO TEC method can be expressed as follows:

$$B_{i} = 0 - {\text{sTEC}}_{\text{dailymin}}^{\text{r}}$$

(A.35)

where ${\text{sTEC}}_{\text{dailymin}}^{\text{r}}$ is the daily minimum of the relative TEC. This simple and fast method will be combined with the weighted LMSQ fitting method. However, the ${\text{sTEC}}_{\text{dailymin}}^{\text{r}}$ might sometimes be an outlier. Thus, the derived $B_{i}$ is not always reliable. Outliers are mainly caused by leveling errors, which are associated with cycle slip, multipath effect, and thermal noise. The DCBs of each receiver are estimated using the measurements corresponding to a 35° satellite elevation mask and a local-time window between 6 and 18 h, avoiding ionospheric scintillation periods.

After the application of the above procedures, the biases are removed from the measured TEC to derive the absolute sTEC (Otsuka et al. 2002).

1.7 Absolute vTEC

The slant TEC (sTEC) depends on the ray path geometry through the ionosphere. To estimate a version of this parameter that does not depend on the elevation angle of the ray path, the equivalent vertical (vTEC) is determined (Ma and Maruyama 2003). Each sTEC value is transformed into a vTEC one by the relationship vTEC = sTEC/S(ε), using the slant factor characterized in expression (A.34). This model is related with the latitude and longitude of the IPP, which are immediately determined, together with the satellite azimuth and elevation angles, from the station and satellite geographic coordinates (El-Gizawy 2003). The vTEC values are expressed in TEC units. The positions of the GPS satellites were determined from the GPS Precise Orbit Files presented in the Standard Product (SP3) format, obtained from the International GNSS Service (IGS). However, the IGS positions of the satellites are reported in 15-min intervals, which are large for some applications. To circumvent this problem, the positions of the satellites are estimated every 15 s through cubic interpolation. It can be shown from Eq. (A.34) that the errors in the slant factor S(ε) due to those in the elevation angle ε are minimal.