Ionospheric and Plasmaspheric Delay Characterization for Lunar Terrestrial GNSS Receivers with the Global Core Plasma Model

Keidai Iiyama; Grace Gao

doi:10.33012/navi.785

Abstract

Recent advancements in lunar positioning, navigation, and timing have demonstrated that global navigation satellite system (GNSS) signals, including weak side-lobe transmissions, can be exploited for lunar spacecraft positioning and timing. Although GNSS-based navigation on the Moon has been recently validated, unmodeled ionospheric and plasmaspheric delays remain a significant source of error, particularly given the unique signal geometry and extended propagation paths. This paper characterizes these delays using the Global Core Plasma Model and a custom low-cost ray-tracing algorithm that iteratively solves for bent signal paths. We simulate first-, second-, and third-order group delays, as well as excess path length from ray bending, for GNSS signals received at both lunar orbit and the lunar south pole under varying solar and geomagnetic conditions. Results show that mean group delays are typically on the order of meters, but can exceed 100 m for rays passing through high-electron-density regions (altitudes below 1000 km with respect to Earth) during high solar activity. We also show that bending delays are generally smaller but non-negligible when the ray passes through high-electron-density regions in the ionosphere. In addition, we quantify the influence of signal frequency, geomagnetic K_p index, and solar R12 index. These results provide quantitative insight into the magnitude and variability of ionospheric and plasmaspheric delays for Earth-Moon GNSS signal paths and help characterize measurement errors affecting GNSS-based navigation at lunar receivers.

Keywords

1 INTRODUCTION

Over the past decade, significant progress has been made toward using terrestrial global navigation satellite system (GNSS) signals for positioning, navigation, and timing (PNT) at the Moon. Studies have long shown that GNSS signals can be tracked at lunar distances, despite being far outside their intended coverage. This concept relies on side-lobe signals and portions of main-lobe signals that spill over beyond Earth (Figure 1), providing a weak but usable navigation source in lunar orbit. Several feasibility analyses (Capuano et al., 2017; Delépaut et al., 2020) have used simulations that combine high-sensitivity GNSS receivers and onboard orbit filters, indicating that continuous GNSS navigation from Earth to the Moon is achievable in principle.

FIGURE 1

GNSS main-lobe and side-lobe signals spilling into cislunar space

Recent on-orbit demonstrations have validated GNSS use near the Moon. In February 2025, the National Aeronautics and Space Administration (NASA) and the Italian Space Agency’s Lunar GNSS Receiver Experiment (LuGRE) receiver on a commercial lander successfully acquired Global Positioning System (GPS) and Galileo signals in lunar orbit, at a distance of 401,000 km from the GNSS satellite (Inside GNSS, 2025; Parker et al., 2022). During a 1-h tracking period, LuGRE attained a navigation fix, achieving a position accuracy of ~1.5 km and a velocity accuracy of ~2 m/s in lunar orbit. In parallel, the European Space Agency is developing the Lunar Pathfinder satellite, which will be equipped with the NaviMoon receiver to test GNSS-based navigation in lunar orbit (Giordano et al., 2022).

Two primary challenges limit GNSS positioning and timing performance at the Moon: the extreme propagation distance produces very weak signals, and the resulting geometry leads to poor dilution of precision. A range of techniques has been proposed to mitigate these limitations, including time-differenced carrier-phase navigation (Iiyama et al., 2024), inter-satellite or differential measurements (Cheung et al., 2024; Delépaut et al., 2024; Iiyama & Gao, 2023), and fusion with complementary sensors (Vila & Gao, 2025). All of these approaches depend critically on minimizing the user equivalent range error (UERE), making accurate modeling of signal delays essential for robust positioning under low-observability conditions.

Among potential error sources, ionospheric and plasmaspheric delays are particularly important. Dual-frequency receivers can cancel most of the ionospheric contribution, but at the cost of increased receiver noise, which can be critical when tracking weak side-lobe signals. Single-frequency terrestrial users typically rely on empirical models such as the Klobuchar model (Klobuchar, 1987) or the Neustrelitz total electron content (TEC) model for Galileo (Hoque et al., 2019), but these models are not directly applicable to lunar users. Unlike a near-vertical path to terrestrial users, signals received by lunar users traverse the ionosphere and plasmasphere horizontally, following long paths dominated by plasmaspheric propagation. Thus, accurate delay prediction requires explicit modeling of electron density in the plasmasphere.

The plasmasphere, a region of cold, diffuse plasma extending from altitudes of roughly 1,000 km to beyond 20,000 km, has been the focus of extensive modeling over the past decade. Although its electron density is lower than that of the ionosphere, it can contribute significantly to the TEC along high-altitude or nighttime GNSS paths (Klimenko et al., 2015). Several empirical models are available, including the Global Core Plasma Model (GCPM) (Gallagher et al., 2000), the International Reference Ionosphere (IRI) extended to the plasmasphere (IRI-Plas) (Sezen et al., 2017; Bilitza et al., 2024), NeQuick (Angrisano et al., 2012), and the Neustrelitz plasmasphere model (NSPM) (Jakowski & Hoque, 2018). The GCPM provides a flexible, density-based description of the plasmasphere; IRI-Plas extends the IRI to GNSS altitudes; NeQuick offers a time-dependent three-dimensional representation of the topside ionosphere and plasmasphere; and the NSPM assimilates ground and satellite data for near-real-time applications.

Previous studies have investigated plasmaspheric delays for GNSS receivers in low-Earth and geosynchronous orbits. For example, Chen et al. (2017) developed a global plasmaspheric electron content model using low-Earth-orbit GNSS measurements, and Matsumoto et al. (2023) applied first-order delay corrections from the GCPM to improve precise orbit determination of a geostationary relay satellite. To the best of our knowledge, however, no prior work has characterized plasmaspheric delays for GNSS signals observed in lunar orbit. Understanding how main-lobe and side-lobe signals are affected under different solar and geomagnetic conditions is critical for designing robust lunar PNT systems.

The objective of this paper is to characterize the plasmaspheric delay experienced by GNSS signals as observed by receivers in lunar orbit and on the lunar surface. We employ the GCPM to simulate first-, second-, and third-order ionospheric and plasmaspheric group delays, along with excess path delays caused by ray bending. The GCPM was selected for its ability to incorporate variable geomagnetic and solar activity conditions.

The main contributions of this work are as follows. First, we propose a computationally efficient ray-tracing algorithm that iteratively determines the bent ray path using the Nelder–Mead algorithm (Nelder & Mead, 1965) without requiring gradient evaluations. Second, we present detailed simulations of first-, second-, and third-order ionospheric and plasmaspheric delays, including bending delays, for GNSS signals received at both the lunar south pole and in lunar orbit. Third, we analyze how these delays vary with signal frequency, geomagnetic activity (K_p index), and solar activity (R12 index). This paper expands upon our earlier conference publication (Iiyama & Gao, 2025) by incorporating updated simulations, including the ray-tracing example presented in Section 2.4.

This paper is structured as follows. In Section 2, we describe the simulation method used to characterize the plasmaspheric delay. In Section 3, we present the simulation results, and in Section 4, we conclude the paper and discuss future work.

2 SIMULATION METHOD TO COMPUTE IONOSPHERIC AND PLASMASPHERIC DELAYS OF GNSS SIGNALS FOR LUNAR USERS

2.1 Ionosphere and Plasmasphere Simulation with GCPM

The GCPM (Gallagher et al., 2000) is an empirical electron density model of the inner magnetosphere that provides both the total electron density and the relative concentrations of major ions. This model integrates region-specific submodels for the ionosphere (based on the IRI 2007 (Bilitza & Reinisch, 2008)), plasmasphere, plasmapause, magnetospheric trough, and polar cap, with smooth transitions enforced by hyperbolic-tangent functions. The electron density is specified as a function of the K_p geomagnetic index, date (to set solar activity parameters for IRI), and position in solar magnetic (SM) coordinates, defined as follows (Matsumoto et al., 2023):

X_SM: X–Z plane contains the Sun and Earth.
Y_SM: Completes the right-handed system.
Z_SM: Aligned with Earth’s geomagnetic pole.

Because the IRI and GCPM primarily model the ionosphere, plasmasphere, and inner magnetospheric cold plasma environment, the present analysis neglects plasma contributions outside this region, where electron densities are much lower.

Figure 2 illustrates the electron density predicted by GCPM v2.4 for January 1, 2025, 12:00 Coordinated Universal Time (UTC) (K_p = 3.0), showing the equatorial (X–Y) and meridional (X–Z) planes.

FIGURE 2

Simulated plasmasphere electron density (log scale) for January 1, 2025, 12:00 UTC, K_p = 3.0, using GCPM v2.4: (left) equatorial (X–Y) plane; (right) meridional (X–Z) plane in SM coordinates

The black region represents the Earth; blue and orange lines indicate GPS and Galileo satellite altitudes.

2.2 Ionospheric and Plasmaspheric Delays

For a right-hand circularly polarized signal, such as a GNSS signal, the phase refractive index n and the group refractive index n_gr are given by the following (Hoque & Jakowski, 2012):

$\begin{aligned} n = 1 - \frac{f_{p}^{2}}{2 f^{2}} - \frac{f_{p}^{2} f_{g} \cos θ}{2 f^{3}} - \frac{f_{p}^{2}}{4 f^{4}} [\frac{f_{p}^{2}}{2} + f_{g}^{2} (1 + \cos^{2} θ)] \end{aligned}$ 1

$\begin{aligned} n_{g r} = 1 + \frac{f_{p}^{2}}{2 f^{2}} + \frac{f_{p}^{2} f_{g} \cos θ}{f^{3}} + \frac{3 f_{p}^{2}}{4 f^{4}} [\frac{f_{p}^{2}}{2} + f_{g}^{2} (1 + \cos^{2} θ)] \end{aligned}$ 2

where $f_{p} = n_{e} e^{2} / (4 π ϵ_{0} m)$ and $f_{g} = e B / (2 π m)$ are the plasma and gyroscope frequencies, respectively. Above, e is the electron charge, ϵ₀ is the permittivity of free space, m is the electron mass, B is the magnetic field strength, f is the frequency of the signal, and θ is the angle between the ray path and the magnetic field line (Hoque & Jakowski, 2011).

The ionospheric and plasmaspheric phase advance d_I and group delay d_Igr of the GNSS signal along the path are given as follows:

$\begin{aligned} d_{I} & = \int (1 - n) d s = \frac{p}{f^{2}} + \frac{q}{2 f^{3}} + \frac{u}{3 f^{4}} \end{aligned}$ 3

$\begin{matrix} d_{I g r} = \int (n_{g r} - 1) d s = \frac{p}{f^{2}} + \frac{q}{f^{3}} + \frac{u}{f^{4}} \end{matrix}$ 4

$\begin{matrix} p = 40.3 \int n_{e} d s = 40.3 T E C = 40.3 (T E C_{L O S} + Δ T E C_{bend}) \end{matrix}$ 5

$\begin{matrix} q = - 2.2566 \times 10^{12} \int n_{e} B \cos θ d s \end{matrix}$ 6

$\begin{matrix} u = 2437 \int n_{e}^{2} d s + 4.74 \times 10^{22} \int n_{e} B^{2} (1 + \cos^{2} θ) d s \end{matrix}$ 7

where ds is the ray path arc and the quantities $d_{I 1} = p / f^{2}, d_{I 2} = q / f^{3}$ , and $d_{I 3} = u / f^{4}$ are the first-order, second-order, and third-order group delays, respectively. The integral of the electron density n_e along the ray path is taken as the TEC. Because the ray path is curved in the ionosphere and plasmasphere, the TEC is split into the line-of-sight (LOS) TEC (TES_LOS) and $Δ T E C_{bend}$ , which is the difference between the TEC along the curved ray path and the LOS between the transmitter and receiver. The first-order delay is the dominant term, whereas the second- and third-order delays are below 1% of the first-order delays, for terrestrial users (Hoque & Jakowski, 2008) and low-Earth orbit receivers (Hoque & Jakowski, 2011).

Ray bending also increases the physical path length, adding an excess path delay:

$d_{l e n} = \int d s - ρ$ 8

where ρ is the geometric distance between the transmitter and the receiver. The procedure for computing the ray path is described in the next section.

For the remainder of this paper, we focus on group delays. The total group delay d_total compared with the LOS distance is then given by the sum of the ionospheric and plasmaspheric group delay and the bending delay:

$d_{t o t a l} = d_{I g r} + d_{l e n} = d_{I 1, LOS} + d_{I 2} + d_{I 3} + d_{I 1, B} + d_{l e n}$ 9

where:

${\begin{cases} d_{I 1, L O S} = 40.3 T E C_{L O S} & First-order LOS group delay \\ d_{I 2} = \frac{q}{f^{3}} & Second-order group delay \\ d_{I 3} = \frac{u}{f^{4}} & Third-order group delay \\ d_{I 1, B} = 40.3 Δ T E C_{bend} & Additional first-order delays due to TEC contributions to ray-path bending \\ d_{l e n} = \int d s - | r_{t x} - r_{r x} | & Excess path length due to bending \end{cases}$ 10

2.3 Ray-Tracing Algorithm

The propagation of the GNSS signal is governed by the vector eikonal equation (Born & Wolf, 1980), as follows:

$\begin{aligned} \frac{d t}{d s} = \frac{n}{c} \end{aligned}$ 11

$\begin{aligned} \frac{d r}{d s} = \hat{s} \end{aligned}$ 12

$\begin{aligned} \frac{d \hat{s}}{d s} = - \frac{\nabla n - \hat{s} (\hat{s} \cdot \nabla n)}{n} \end{aligned}$ 13

where s is the arc length, t is the time, r is the position vector, c is the speed of light, $\hat{s}$ is the unit vector in the direction of propagation, and n is the phase refractive index.

In addition to the differential equations above, the trajectory of the ray is constrained by the boundary conditions at the transmitter and receiver (Caruso et al., 2023):

$\begin{array}{l} t (s = 0) = t_{t x} & r (t_{t x}) = r_{t x} \end{array}$ 14

$\begin{array}{l} t (s = s_{f}) = t_{r x} & r (t_{r x}) = r_{r x} \end{array}$ 15

Note that given the reception time and position of the receiver, the time and position of the transmitter can be solved by iteratively solving the following light-time equation:

$t_{t x} = t_{r x} - \frac{‖ r_{r x} - r_{t x} (t_{t x}) ‖}{c}$ 16

where r_tx(t_tx) is the position of the transmitter at time t_tx. We ran the iteration until the change in the propagation time estimate fell below the numerical tolerance of 10⁻¹² s.

To compute the ray path, we must solve for the unknown initial ray direction ${\hat{s}}_{0}$ and ray length s_f that satisfy the boundary conditions. Several methods have been proposed for solving this problem, such as the shooting method (homing-in approach) (Jones & Stephenson, 1975; Strangeways, 2000) and variational methods (Coleman, 2011).

Here, we solve the ray-tracing problem using a shooting method accounting for the geometry of the ray path between the GPS satellite and the lunar receiver, as illustrated in Figure 3.

FIGURE 3

The ray-tracing problem for GNSS signals received by a lunar user

The ray path is divided into the plasmasphere section and the vacuum section, based on the observation that the electron density is significantly lower in higher-altitude regions past the plasmapause. In this paper, we assume that the ray path is straight once it passes the cutoff radius of 4 RE.

We first divide the ray path into the plasmasphere section and the vacuum section, based on the observation that the electron density is significantly lower in higher-altitude regions above 2–3 Earth radii (RE) (Gallagher et al., 2000). We assume that the ray path is straight once it exists beyond this boundary (set as 4 RE from Earth’s center in this work). With this assumption, the s_f that minimizes the distance between the propagated ray and the receiver can be computed as follows:

$\begin{array}{l} s_{f} & = & s_{e x i t} + Δ s \\ Δ s & = & (x_{r x} - x_{e x i t}) \cdot {\hat{s}}_{e x i t} \end{array}$ 17

where s_exit, x_exit, and ${\hat{s}}_{exit}$ are the arc length, position, and unit vector of the ray at the exit point of the plasmasphere, which can be computed by integrating the ray-tracing equations from the transmitter to the exit point.

With this assumption, the ray-tracing problem can be solved by searching for the initial ray direction ${\hat{s}}_{0}$ that satisfies the boundary conditions. Because it is expensive to evaluate the electron density required for the evaluation of $d \hat{s} / d s$ , we apply the following correction to correct for the initial direction ${\hat{s}}_{0}$ :

Compute the ray path from the transmitter until it reaches the cutoff radius (4 RE) by integrating the ray-tracing equations using the fourth-order Runge–Kutta method at discrete points $s_{0}, s_{1}, \dots, s_{exit}$ along the ray. We compute the ray path by integrating Equation (13). We apply an adaptive step size based on the altitude of the ray, where we use Δs = 10 km when the altitude is below 1000 km, Δs = 20 km when the altitude is below 4000 km, and Δs = 100 km otherwise. We store $d \hat{s} / d s$ at each point.
Terminate integration once the ray reaches the cutoff radius. Evaluate the final position x_rx of the ray at the receiver using the following equation:
${\overset{―}{x}}_{r x} = x_{e x i t} + Δ s {\hat{s}}_{e x i t}$ 18
Run the Nelder–Mead optimization algorithm (Nelder & Mead, 1965) to compute the initial ray direction that minimizes the distance between the propagated final position ${\overset{―}{x}}_{r x}$ and the receiver position x_rx. It has been found that the Nelder–Mead simplex algorithm works reliably when solving for the initial ray direction ${\hat{s}}_{0}$ (Hoque & Jakowski, 2008). Instead of directly optimizing for the direction vector ${\hat{s}}_{0}$ , we optimize for the azimuth and elevation angles of the initial ray direction, which corresponds to a twodimensional optimization problem. In the propagation for the optimization, we use the stored $d \hat{s} / d s$ at each point to compute the ray path, instead of evaluating the electron density at each point, to avoid calling the GCPM at each iteration.
Repeat steps 1–3 until the distance between the propagated final position ${\overset{―}{x}}_{r x}$ and the receiver position x_rx falls below a specified threshold or until the terminal position error stops decreasing.

Without any correction, the final position error is on the order of 10-100 km, due to the bending of the ray path in the plasmasphere. By applying the correction, the difference between the propagated final position ${\overset{―}{x}}_{r x}$ and the receiver position x_rx can be reduced to within sub-meter to 100 m, ensuring convergence of the computed total ionospheric and plasmaspheric delay as described in the next section.

2.4 Ray-Tracing Example

To illustrate the performance of the proposed ray-tracing algorithm, this section presents an example ray-tracing result between a transmitter (pseudorandom noise [PRN] 4 GPS satellite) located at r_tx = [24513.42 1876.09 10266.99] km and a lunar receiver located at r_rx = [−343532.59 −125200.76 −123527.20] km in the J2000 frame. The tangential altitude of the ray is 163.5 km, passing through the ionosphere.

Figure 4 shows the reduction in terminal position error and group-delay difference over iterations. The L1 signal converged to a terminal error of 1.27 m in five iterations, whereas the lower-frequency L5 signal converged to 12.89 m in six iterations. First-order and bending-delay errors decreased below 1 mm after three iterations, when the terminal position error was already ~22 m. Because the L1 signal experiences less bending, it converges faster and with a smaller final position error.

FIGURE 4

Terminal position error (top) and differences in the converged delay value (bottom) over iterations

The L1 signal converges faster than the L5 signal, which experiences larger delays. Three iterations lead to a delay simulator error below 1 mm.

Figure 5 displays the displacement between the bent ray path and the straight transmitter-receiver line. The largest offset occurs near the point of lowest altitude, with the L5 signal exhibiting greater bending due to its lower frequency and correspondingly smaller refractive index.

FIGURE 5

Distance between the ray path and the straight line connecting the transmitter and the receiver

The distance to the straight line reaches a maximum near the lowest altitude point. The L5 signal experiences greater bending than the L1 signal.

3 SIMULATION RESULTS

3.1 Simulation Configuration

3.1.1 Lunar Satellites

We simulated five users in the lunar orbit, corresponding to NASA’s Lunar Communications Relay and Navigation Systems (LCRNS) satellites, with the initial orbital elements as shown in Table 1. The initial states were taken from the reference constellation (Ryden & Volle, 2025), and the initial epoch was set to March 1, 2027. The propagated LCRNS satellite orbits are shown in Figure 6. We also simulated GPS tracking for a user at the lunar south pole, resulting in a total of six users in the simulation.

View this table:

TABLE 1 Initial Condition of the LCRNS Satellites

FIGURE 6

Orbits of the five LCRNS satellites in a Moon-centered inertial frame

3.1.2 GNSS Satellites and Tracking Simulations

The orbits of the GPS and Galileo satellites were initialized using the SP3 files provided for the epoch June 23, 2025. The GPS and Galileo satellite constellation with the lunar direction for the initial epoch is shown in Figure 7.

FIGURE 7

GPS (blue) and Galileo (orange) satellite constellation with the lunar direction for the initial epoch (2027/03/01, 12:00:00 UTC)

A fixed transmission gain of 14.3 dBi was assumed for all GNSS satellites, and GNSS antenna patterns were initialized using the references available online. We used the parameters from the NASA antenna characterization experiment (Donaldson et al., 2020) for Block II-F, the original data by Lockheed Martin for Block IIR and IIR-M satellites (Marquis & Reigh, 2015), the United States Coast Guard Navigation Center (Fischer, 2022) for Block-III, and the Publications Office of the European Union (Menzione et al., 2024) for the equivalent isotropically radiated power of the Galileo satellites. The antenna patterns of the selected GPS and Galileo satellites and the GNSS receiver assumed for the lunar users are shown in Figure 8.

FIGURE 8

Antenna patterns of selected GPS and Galileo satellites and the lunar GNSS receiver

We assumed that the GNSS receivers of the lunar users have a peak gain of 14 dBi, with a half-power beamwidth of 6^∘. We also assumed that the GNSS receiver is able to track the signals from the GPS and Galileo satellites with a C/N0 of 18 dB-Hz or higher, based on experimental results for the NaviMoon receiver (SpacePNT, 2024). The receiver noise errors of the pseudorange measurements are simulated assuming thermal noise in the delay-locked loop (DLL), which depends on the carrier-to-noise ratio of the received signal (Kaplan & Hegarty, 2017).

3.1.3 Software

For simulation, we created a simulation software that interfaces the GCPM v2.4 model (implemented in Fortran) with the open-source lunar PNT library LuPNT (Iiyama et al., 2023; Vila et al., 2025), which was developed in Python and C++. Several modifications were introduced to the original GCPM code to compute electron density values for recent epochs (from 2015 onward) and to accept custom solar activity parameters (R12 index) as inputs. The simulation software computes the GNSS and lunar satellite orbits using LuPNT and performs ray tracing by using the GCPM to compute electron density values along the ray path. This software is available at https://github.com/kdricemt/pecsim.

3.2 Ionospheric and Plasmaspheric Delay Simulation Results

3.2.1 Results for the Baseline Test Case

The baseline scenario was simulated using electron densities for January 1, 2025, 12:00 UTC, with a simulation span of 45 h—approximately one orbital period of the LCRNS satellites. This epoch was selected because the GCPM cannot be run for the target date of March 1, 2027, owing to unavailable historical parameters. To represent moderate geomagnetic activity, we set the K_p index to 3.0 (the actual value at the epoch was 6.667). For solar activity, we used historical measurements of R12 = 167.24 and F10.7 = 210.39, corresponding to the high solar activity expected near the peak of Solar Cycle 25. As discussed in Sections 3.2.2 and 3.2.3, the resulting delays are sensitive to both geomagnetic and solar activity levels.

Simulation outputs for GPS L1 and Galileo E1 signals are summarized in Table 2 and illustrated in Figures 9 and 10. The results include all signals received by the LCRNS satellites and a lunar south-pole user, sampled every 30 min along the orbits.

View this table:

TABLE 2 Mean Group Delays and C/N0 Values Observed at Lunar Receivers for the Baseline Scenario (K_p = 3.0, R12 = 167.24, 2025/01/01 12:00:00 UTC)

See Figure 3 for a definition of the tangential altitude. The second row indicates the ratio of signal that the receiver could use if all signals below the lower range of the tangential altitude band are eliminated (e.g., <1000 km for range of 1000-2000 km).

FIGURE 9

Histogram of ionospheric and plasmaspheric delays and related parameters for the baseline scenario (K_p= 3.0, R12 = 167.24, 2025/01/01 12:00:00 UTC)

Scatterplot of ionospheric and plasmaspheric delays for the nominal scenario Both the group delay and C/N0 decrease as the tangential altitude increases, creating a trade-off between C/N0 and group delay (Kp= 3.0, R12 = 167.24, 2025/01/01 12:00:00 UTC).

FIGURE 10

Scatterplot of ionospheric and plasmaspheric delays for the nominal scenario Both the group delay and C/N₀ decrease as the tangential altitude increases, creating a trade-off between C/N₀ and group delay (K_p= 3.0, R12 = 167.24, 2025/01/01 12:00:00 UTC).

Figure 9 shows the distributions of total delay, bending delay, tangential altitude, C/N₀, receiver range error, and boresight angle for the baseline case. The total delay is broadly distributed, with most samples ranging from sub-meter to several meters and a smaller number of signals exhibiting much larger delays. In contrast, the bending delay is several orders of magnitude smaller. We also observe that the majority of the received signals are from the side lobes (boresight angle $⪆ 20^{\circ}$ ), with a C/N₀ below 25 dB-Hz, which generates pseudorange errors greater than 20 m.

Table 2 and Figure 10 show that ionospheric and plasmaspheric delays become significant for low-altitude ray paths. For tangential altitudes below 500 km, the mean first-order group delay reaches up to 50.59 m, but the delay decreases rapidly with altitude: above 10,000 km, the mean group delay falls below 0.2 m. Bending delays are much smaller than the first-order term but can still exceed 1 m for paths with tangential altitudes under 1000 km. Notably, the additional delay caused by bending exceeds the actual increase in geometric path length. Second- and third-order delays remain negligible, several orders of magnitude smaller than the first-order contribution.

The carrier-to-noise ratio (C/N₀) values reported in Table 2 further highlight a bias-variance trade-off. Signals with low tangential altitudes experience larger ionospheric and plasmaspheric delays (higher bias) but benefit from a higher C/N₀ and thus lower receiver thermal noise (lower variance). This trend arises because the low-tangential-altitude signals are transmitted at smaller angular offsets from the transmitter antenna boresight, resulting in a higher transmit antenna gain. Conversely, high-altitude signals exhibit smaller propagation delays but a lower C/N₀, resulting in higher receiver noise. This interplay between delay and noise is a key consideration when using GNSS signals in lunar orbits.

3.2.2 Effect of Geomagnetic Activity (K_p Index)

To analyze the effect of geomagnetic activity on ionospheric and plasmaspheric delays, we simulated the electron density for five different K_p indices, K_p = 1,3,5,6.667, and 9.0. The electron density at the X–Y (equatorial) and X–Z (meridian) planes for different K_p indices is shown in Figure 11. We observe that as the K_p index increases, the plasmapause (the sharp boundary between the dense plasmasphere and the lower-density plasmatrough outside it) contracts from 4–6 RE under quiet conditions to 2–3 RE under strong geomagnetic activity, reflecting erosion of the plasmasphere.

FIGURE 11

Electron density in the X–Y (top row) and X–Z (bottom row) plane for different K_p indices

The epoch is set to January 1, 2025, 12:00:00 UTC, and R12 is set to 167.24 (value at the epoch). The plasmapause shrinks as the K_p index increases, owing to erosion of the plasmasphere.

Figure 12 presents histograms of the ionospheric and plasmaspheric delays observed by LCRNS-1 and the lunar south-pole user across the different K_p levels. Consistent with the plasmapause contraction, higher K_p indices generally produce smaller group delays, as the region of high electron density is reduced.

FIGURE 12

Histogram of ionospheric and plasmaspheric delays for different K_p indices The delay decreases as the K_p index increases, owing to shrinkage of the plasmapause.

3.2.3 Effect of Solar Activity (R12 Index)

To assess the influence of solar activity on ionospheric and plasmaspheric delays, we simulated electron densities for five levels of the R12 index: R12 = 10, 50, 100, 150, and 200. R12 is the 12-month smoothed sunspot number and serves as a standard indicator of solar activity. In the GCPM, the IRI submodel requires the F10.7 solar flux and IG12 ionospheric indices, which we derived from R12 using the empirical Provision of High-frequency Raytracing Laboratory for Propagation studies (PHaRLAP) relationships (Cervera, 2025):

$\begin{aligned} F10.7 = 63.75 + 0.728 \times R12 + 0.00089 \times R12^{2} \end{aligned}$ 19

$\begin{aligned} IG12 = - 12.349154 + 1.4683266 \times R12 - 0.00267690893 \times R 12^{2} \end{aligned}$ 20

Electron density distributions for each R12 level in the X–Y (equatorial) and X–Z (meridional) planes are shown in Figure 13. Corresponding histograms of ionospheric and plasmaspheric delays observed by LCRNS-1 and the lunar south-pole user are shown in Figure 14. As expected, higher R12 values produce larger group delays, reflecting the increase in ionospheric and plasmaspheric electron density during periods of elevated solar activity.

FIGURE 13

Electron density in the X–Y (top row) and X–Z (bottom row) plane for different R12 indices

The epoch is set to January 1, 2025, 12:00:00 UTC, and the K_p index is set to 3 for all cases. The electron density is shown on a log scale.

FIGURE 14

Histogram of ionospheric and plasmaspheric delays for different R12 indices The delay becomes larger as the R12 index increases, owing to the increase in electron density in the ionosphere and plasmasphere.

Table 3 summarizes the mean and 99th percentile group delays across R12 levels and tangential altitudes. These results provide practical guidance for setting altitude masks to exclude signals likely to contain extensive group delays. For instance, to maintain the 99% group delay below 5 m, an altitude mask of approximately 2000 km suffices when R12 = 50, but the threshold must increase to 3000 km for 100 ≤ R 12 ≤ 150 and to 10,000 km when R12 = 200. This scenario corresponds to removing approximately 8.7% of the rays for R12 = 50 and 57.5% for R12 = 200, as shown in Table 2. Thus, during periods of high solar activity, ionosphere-free combinations are desirable to avoid a significant loss of usable signals.

View this table:

TABLE 3 Mean and 99th Percentile Total Group Delays for Different R12 Values and Tangential Altitudes (L1 Signal, K_p= 3.0, 2025/01/01 12:00:00 UTC)

See Figure 3 for a definition of the tangential altitude.

3.2.4 Simulation Results for Different Frequencies

We evaluated the effect of signal frequency by simulating the group delays and receiver thermal noise for two civilian GPS signals: L1 (1575.42 MHz) and L5 (1176.45 MHz). The results are summarized in Table 4 and Figure 15.

View this table:

TABLE 4 Ionospheric and Plasmaspheric Group Delays for L1 and L5 Signals

The error statistics are for the epoch 2025/01/01 12:00:00 UTC, R12 = 167.24, K_p index = 3.0.

FIGURE 15

Mean total ionospheric and plasmaspheric group delay, receiver noise, and total UERE (=receiver noise + ionospheric delay) for GPS L1 and L5 signals with different tangential altitude ranges

The results are for the epoch 2025/01/01 12:00:00 UTC, K_p index = 3.0, and R12 index = 167.24. The total UERE for L1 is smaller than that of L5 for altitudes below 2000 km, owing to the larger ionospheric and plasmaspheric delay added to the L5 signal.

The simulations show that the signal preference depends strongly on the tangential altitude of the ray path. For rays with tangential altitudes below approximately 2000 km, the ionospheric and plasmaspheric group delays dominate the error budget, exceeding the receiver noise by one to three orders of magnitude. Across all altitudes, the mean group delays of the L1 signal are approximately 53%–55% of those of L5, whereas the L5 receiver noise is only 7%–8% of the L1 noise. This result presents a clear trade-off: L1 offers a smaller bias from group delay, whereas L5 provides a lower variance from thermal noise.

To quantify the combined effect, we computed the total UERE, defined as the mean absolute sum of the group delay and pseudorange noise over 100 samples:

$Total UERE = \frac{1}{N_{s}} \sum_{i = 1}^{N_{s}} | d_{total} + ϵ_{ρ} |$ 21

where N_s = 100 and $ϵ_{ρ} \sim N (0, σ_{ρ})$ represents DLL thermal noise. The noise variance $σ_{ρ}^{2}$ is derived from the carrier-to-noise ratio (C/N₀) using the following expressions (Kaplan & Hegarty, 2005):

$σ_{ρ}^{2} = {\begin{cases} \frac{B_{DLL}}{2 (C / N_{0})}, d (1 + \frac{2}{T (C / N_{0}) (2 - d)}), & d \geq \frac{π}{T_{c} B_{fe}} \\ \frac{B_{DLL}}{2 (C / N_{0})} (\frac{1}{T_{c} B_{fe}}) (1 + \frac{1}{T (C / N_{0})}), & d \leq \frac{π}{T_{c} B_{fe}} \\ \frac{B_{DLL}}{2 (C / N_{0})} (\frac{1}{T_{c} B_{fe}} + \frac{B_{fe} T_{c}}{π - 1} {(d - \frac{1}{T_{c} B_{fe}})}^{2}) (1 + \frac{2}{T (C / N_{0}) (2 - d)}), & \frac{1}{T_{c} B_{fe}} < d < \frac{π}{T_{c} B_{fe}} \end{cases}$ 22

Parameter definitions and values are listed in Table 5. Here, we used the parameters employed in the preliminary analysis for the LCRNS satellites (Mina et al., 2025). Following their receiver parameters, for this section only, the C/N₀ acquisition threshold was set to 22 dB-Hz.

View this table:

TABLE 5 Parameters for DLL Thermal Noise Calculation (Mina et al., 2025)

As shown in Table 4 and Figure 15, L1 achieves a smaller total UERE than L5 for tangential altitudes up to roughly 2000 km because its smaller group delay outweighs its higher receiver noise. Conversely, L5 becomes advantageous for side-lobe paths where group delays are intrinsically small and thermal noise dominates. Future work will examine optimal combinations of ionosphere-free dual-frequency measurements and single-frequency tracking to minimize the total error across different altitude regimes.

4 CONCLUSION

This study presented a detailed simulation framework for evaluating ionospheric and plasmaspheric delays affecting GNSS signals received at the Moon. Using the GCPM and a custom ray-tracing algorithm, we quantified first-order, higher-order, and bending delays for both orbital and surface users across a wide range of geomagnetic and solar activity levels.

The results show that mean group delays are generally near 1 m but can reach beyond 100 m for low-altitude ray paths, particularly during periods of high solar activity. Delays increase with increasing R12 index and decrease with higher geomagnetic K_p index, reflecting the contraction of the plasmapause and the corresponding changes in electron density. Signal frequency further influences the trade-off between bias and variance: L1 signals experience smaller group delays but higher receiver noise, whereas L5 signals exhibit the opposite behavior, leading to altitude-dependent performance differences.

Future work will focus on validating these simulations with upcoming LuGRE measurements, exploring mitigation techniques such as ionosphere-free combinations and single-frequency optimization, and integrating delay estimation directly into navigation filters for lunar satellites. In addition, it will be important to develop data assimilation methods to improve plasmaspheric density modeling using observed GNSS side-lobe measurements from lunar receivers, as well as lightweight delay-approximation methods suitable for onboard implementation. Because models such as the GCPM are empirical in nature, these efforts are expected to improve the characterization of real-time spatial and temporal variations in plasmaspheric electron density under varying solar and geomagnetic conditions. By quantifying the magnitude and variability of ionospheric and plasmaspheric propagation effects, this work provides a foundation for improved error characterization and mitigation in GNSS-based positioning and timing for cislunar and lunar applications.

HOW TO CITE THIS ARTICLE:

Iiyama, K., & Gao, G. (2026). Ionospheric and plasmaspheric delay characterization for lunar terrestrial GNSS receivers with the global core plasma model. NAVIGATION, 73. https://doi.org/10.33012/navi.785

Acknowledgments

This material is based upon work supported by the Nakajima Foundation.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Ionospheric and Plasmaspheric Delay Characterization for Lunar Terrestrial GNSS Receivers with the Global Core Plasma Model

Abstract

1 INTRODUCTION

2 SIMULATION METHOD TO COMPUTE IONOSPHERIC AND PLASMASPHERIC DELAYS OF GNSS SIGNALS FOR LUNAR USERS