Research ArticleOriginal Article
Open Access

Optimal INS Monitor for GNSS Spoofer Tracking Error Detection

Birendra Kujur, Samer Khanafseh, and Boris Pervan
NAVIGATION: Journal of the Institute of Navigation March 2024, 71 (1) navi.629; DOI: https://doi.org/10.33012/navi.629

Abstract

In this article, we describe a new method for detecting global navigation satellite system (GNSS) spoofing using an inertial navigation system. We specifically address the most difficult-to-detect scenario, in which a spoofer replicates the authentic GNSS signal with only additive errors due to the spoofer’s uncertainty in knowledge of the target’s position. We derive an optimal monitor to detect the anomalous temporal structure of the spoofed measurements caused by the spoofer’s target tracking errors. This new monitor uses accumulated Kalman filter innovations projected into the position state domain. We demonstrate how the monitor window length can be set to achieve any required missed detection probability, and we evaluate the performance of the monitor for both white and colored tracking error. Finally, we present a complementary solution separation monitoring concept to detect rapid-onset spoofing and to achieve protection levels in real time.

Keywords

1 INTRODUCTION

The civil infrastructures behind safety-critical applications in aviation, maritime, and terrestrial navigation rely heavily on global navigation satellite systems (GNSSs). The civil GNSS signal structures are publicly known and vulnerable to spoofing attacks, which endangers public safety (Humphreys et al., 2008). In an attack, the spoofer feeds a counterfeit signal to the targeted user to cause faulty position or time estimates. Various spoofing attack methods and respective defense techniques have been summarized by Jafarnia-Jahromi et al. (2012b), Gunther (2014), Psiaki & Humphreys (2016), and Fernández-Hernandez et al. (2019). Spoofing signals can be created in several ways, for example, by transmitting self-consistent synthetic GNSS signals using a GNSS simulator, by meaconing (i.e., recording authentic GNSS signals and rebroadcasting them with a delay), or by recording and replaying authentic signals with altered navigation data, among others. Spoofed signals can be sent to target receivers via a single transmit antenna or an array of multiple antennas, static or mobile, to mimic the satellite line-of-sight geometries. A spoofer may choose to initially send the counterfeit signals at a lower power than the authentic signals and then slowly increase the power to cause the target receiver’s tracking loops to transition from the authentic to the spoofed signals and to then slowly drag the user along a false trajectory. A spoofer can also opt to jam the GNSS signals before transmitting the spoofed signal. A more sophisticated spoofer could send two spoofed signals, where one of the spoofed signals would be in opposite phase to the authentic signals, causing cancellation. In this process, known as nulling, the target receiver would not even be able to detect multiple correlation peaks. Ideally, spoofing detection techniques should protect against all possible scenarios.

Potential detection techniques include signal-processing methods, such as power and distortion monitoring (Turner et al., 2020; Wesson et al., 2018), cryptographic authentication (Kerns, Wesson, et al., 2014; Wesson et al., 2011), correlation comparison with encrypted authentic signals (O’Hanlon et al., 2013), correlation peak comparison within a receiver (Rothmaier et al., 2021b), combining different observables (Broumandan et al., 2020; Rothmaier et al., 2021a), use of multiple receivers (Stenberg et al., 2020), spoofing discrimination using spatial processing by antenna arrays (Nielsen et al., 2014), GNSS signal direction of arrival comparison (Meurer et al., 2012), code and phase rate consistency checks (Moshavi, 1996), high-frequency antenna motion (Psiaki et al., 2013), automatic gain control schemes (Akos, 2012), and signal power monitoring techniques (Jafarnia-Jahromi et al., 2012a). Some of these methods are indeed effective, but they have various computational, logistical, and physical limitations. Augmenting data from auxiliary sensors such as inertial measurement units (IMUs), barometric altimeters, and independent radar sensors to discriminate spoofing has also been proposed (Kerns, Shepard, et al., 2014; Lo et al., 2017; Swaszek et al., 2016).

Our group introduced the first stochastic description and quantification of the performance of an IMU-based GNSS spoofing monitor against worst-case “faults” (i.e., spoofing inputs over time) (Khanafseh et al., 2014; Tanil, Khanafseh, et al., 2018; Tanil et al., 2015a, 2015b; Tanil, Khanafseh, et al., 2016; Tanil et al., 2017). We specifically investigated anti-spoofing solutions utilizing IMUs because essentially all modern aircraft are equipped with IMUs, thereby requiring minimal additional cost or system modification. An IMU is naturally immune to external interference, which makes it an excellent resource for ensuring navigation continuity. Additionally, when used in the navigation solution in various integration schemes with GNSS (uncoupled or loosely, tightly, or ultra-tightly coupled), the INS provides the redundancy needed to resist spoofing attacks. In our prior work (Tanil, Khanafseh, et al., 2018; Tanil et al., 2017), we developed a chi-squared innovation sequence detector to monitor the accumulated time history of normalized Kalman filter (KF) innovations. The two main advantages of this cumulative innovation (CI) sequence monitor are that innovations are already available in the KF, such that little additional computation is required for the monitor implementation, and that it provides detection capability against slowly growing faults. We evaluated the performance of the CI monitor against worst-case GNSS fault profiles both analytically and experimentally (Tanil, Jimenez, et al., 2018; Tanil, Khanafseh, et al., 2018). The worst-case fault here represents a spoofed GNSS signal profile that maximizes integrity risk. We also analyzed the sensitivity of the CI monitor against error modeling uncertainties in the INS/GNSS KF structure (Kujur et al., 2019).

However, post-detection recovery has not been addressed in previous work— the difficulty being that the KF is already corrupted once spoofing is detected. Moreover, previous performance evaluations assumed that the CI monitor started at spoofing onset and operated without a defined run time. Finally, the CI monitor did not provide the means to produce a protection level—i.e., a position-domain containment boundary corresponding to the maximum acceptable level of integrity risk. To address these critical limitations, in this paper, we introduce a new type of CI monitor: a cumulative position-domain innovation (CPI) monitor that detects spoofing by accumulating the target position tracking error embedded in the spoofer’s signal. We also present a complementary solution separation (SS) concept to produce protection levels and provide a means for post-detection recovery.

Section 2 provides useful background information on the original CI monitor. Section 3 introduces the new CPI monitor. An example application with quantitative results is presented in Section 4. In Section 5, we present the complementary SS monitor. Finally, we summarize our work in Section 6. We also provide relevant derivations in the appendices.

2 KF STATE MODEL

We consider a vehicle employing INS and GNSS sensors integrated with a KF to estimate its position, velocity, and attitude. The dynamics of the INS/GNSS system, augmented as needed with sensor error state dynamics, are linearized to obtain the process model utilized in the KF:

Embedded Image 1

where xk is the state vector, Φk is the state transition matrix, Embedded Image is the process noise model matrix, and wk is the additive white process noise with a respective covariance matrix Qk. The measurement model is as follows:

Embedded Image 2

where Hk is the observation matrix and vk is the measurement noise with a respective covariance matrix Vk.

The innovation vector γk at time epoch k is defined as follows:

Embedded Image 3

where Embedded Image is the state vector estimate prior to the measurement update at time epoch k. The CI detector is a chi-squared monitor that utilizes the cumulative normalized innovations from a KF as the test statistic and compares this test statistic against a threshold. A cumulative test statistic qN at time epoch N is the sum of squares of the normalized innovation vectors over time, given as follows:

Embedded Image 4

where Embedded Image is the innovation vector covariance matrix at time epoch k and Embedded Image is the estimate error covariance matrix prior to the measurement update at time epoch k. The monitor simply checks whether the test statistic qN is smaller than a predefined threshold Embedded Image. For a given false alarm requirement under fault-free conditions, the threshold TN is determined from the inverse chi-squared cumulative distribution function (CDF) with N degrees of freedom. The monitor produces an alarm if qN > TN.

One of the major limitations of the CI monitor is that it does not offer a means for post-detection recovery because there is no fault-free source upon which to rely. Another unaddressed issue concerns the monitor start and run times. In all prior performance evaluations, it was assumed that the monitor start time (conveniently) coincided with the spoofing onset time, and no guidance was provided to determine how long the monitor should run before resetting. In the next section, we introduce the CPI monitor and show that it directly addresses the run time issue and also provides superior detection performance relative to the CI monitor for any run time. The start time and recovery are addressed afterward.

3 CPI MONITOR

3.1 Spoofer’s Tracking Error

The initial objective of a smart spoofer is to cause the target receiver to lose lock of the authentic GNSS signals and lock onto the counterfeit signals without being detected. This initial lock transition is crucial because any abrupt changes during the process would be easily detectable, sabotaging the spoofer’s plan. The best way for a spoofer to cause the target to switch the counterfeit signal without being detected is to initially replicate the authentic signals at lower power and then slowly increase the power to cause the target receiver to transition. If successful, the spoofer would then attempt to inject small but accumulating position and/or time offsets in an attempt to slowly pull the target away along the desired spoofed trajectory while remaining undetected.

To deliver a replica of the authentic signal to the target, the spoofer would need to know the position of the target’s GNSS antenna during the attempted takeover. Consequently, the spoofer would need to track the target in real time. Any tracking errors would ultimately be embedded in the replica signal and appear to the target as additional “noise” in the received GNSS signals. As tracking errors are inevitable, detecting the presence of such unusual noise would expose active spoofing. This is true even during the initial takeover phase when the spoofer has not yet injected any additional offsets to the spoofed signals. In the remainder of this paper, we address this case specifically, as any additional offsets to the spoofing profile are irrelevant to the monitor. The spoofer’s uncertainty of the target position will simply be referred to as “tracking error” in the remainder of the paper. In Appendix C, we show that the tracking error appears in the spoofed measurements as an additive term, and the measurement equation can be represented as follows:

Embedded Image 5

where Embedded Image is the spoofed measurement vector, zk is the “true” (unspoofed) measurement vector, Hk is the observation matrix that maps the tracking error to the range domain, and Embedded Image is an n × 1 column vector of tracking error, with n being the number of states. The superscript s indicates “spoofed.”

We can observe these tracking errors in the KF innovation vector because it extracts the difference between the GNSS measurements and the predicted measurements of the process model from the INS. We will see later that even small deviations over time are readily observable when the INS is used because of the precision of GNSS carrier phase measurements.

Because the innovation vector γk in Equation (3) is constructed from the entire measurement vector zk and the entire state vector Embedded Image, the effect of the position tracking errors, our proxy for spoofing, will not be the only contributor to γk. A more direct way to observe the effect of the tracking errors would be to project the innovation vector into the position domain.

3.2 Neyman–Pearson Optimal and Sufficient Test Statistic

We desire to find a test statistic to maximize the probability of detection for a given false alarm rate. The Neyman–Pearson lemma (Neyman & Pearson, 1933) provides the solution to this problem: given two mutually exclusive hypotheses H0 and H1 which for some observation x have conditional probability densities p0(x | H0) and p1(x | H1), the likelihood ratio is the optimal test statistic. The likelihood ratio is defined as follows:

Embedded Image 6

Our two hypotheses are the fault-free case, which has no tracking error, and the spoofed case, in which the tracking error appears in the innovation vector. As a starting point, we model the tracking error Embedded Image as white Gaussian noise (WGN) distributed as Embedded Image, where Embedded Image is the unknown variance of the tracking error.

We consider the innovation vector distributions with and without the tracking error for our likelihood ratio test. The fault-free innovation vector distribution at any time k is given as Embedded Image, whereas the spoofed innovation vector distribution, as shown in Appendix C, is given as Embedded Image, where R = uuTσt and u is a unit vector in an arbitrary spatial direction in which the tracking error exists. The innovation vectors are mutually independent over time under H0. In principle, the vectors can also be modeled as independent under H1 given that a “high-quality” IMU is being utilized because Embedded Image in Equation (2) would be relatively unaffected by the spoofer’s tracking error if a high-quality IMU were used. (Note: We refer to IMUs with specifications equivalent to or better than the navigation-grade IMU listed in Appendix F as “high-quality” and IMUs with specifications equivalent to or lesser than the automotive-grade IMU listed in Appendix F as “low-quality/grade.”) We accept this assumption as true in the following derivation of the optimal test statistic, and in Appendix D, we show that this assumption is valid even for small tracking errors and “low-grade” IMUs.

Using Equation (6) and given an arbitrary threshold λ(N) for independent innovations γk(k = 1,…, N), the likelihood ratio test can be expressed as follows (Kay, 1998):

Embedded Image 7

where:

Embedded Image 8

captures the terms that do not depend on γk(k = 1,…, N). Taking the natural log of both sides and collecting all of the constant terms on the right, we obtain the following:

Embedded Image 9

Moreover, because the matrices Sk and Embedded Image are non-singular, we can write the following (Miller, 1981):

Embedded Image 10

Substituting Equation (10) into Equation (9), we obtain the following relation:

Embedded Image 11

where the constant on the right side is as follows:

Embedded Image 12

For short periods of time, matrices H and S can be assumed to be time-invariant; thus, the term Embedded Image can be moved to the right side of Equation (11). Equation (11) then reduces as follows:

Embedded Image 13

where:

Embedded Image 14

Recalling that R = uuTσt, we substitute this term into Equation (13) to obtain the following:

Embedded Image 15

Rearranging the terms of Equation (15) and noting that S−1 is symmetric, we can now write the following relation:

Embedded Image 16

where we have defined C4(N) = C3(N)/σt.

Finally, we introduce the following scalar projection of the innovation vector:

Embedded Image 17

which is a weighted projection of the innovation vector into the position-domain direction u, the tracking error direction under consideration. Therefore, the optimal test statistic is given as follows:

Embedded Image 18

which we call the CPI. We note that Embedded Image does not require a knowledge of σt.

3.3 Position-Domain Innovation

Under spoof-free conditions, the scalar position-domain innovation in Equation (17) is distributed as follows:

Embedded Image 19

To simplify the notation, we define the variance as follows:

Embedded Image 20

When the spoofer sends a signal mimicking the authentic signal with some inherent additive tracking error, the latter can be observed in the position-domain innovation. In Appendix C, we show that the innovation for the spoofed case with tracking error for an arbitrary spatial direction u is given by the following:

Embedded Image 21

Substituting Equation (21) for σk in Equation (17), we obtain the spoofed position-domain innovation:

Embedded Image 22

The mean of Embedded Image is again zero, but the variance is now as follows:

Embedded Image 23

This can be written more compactly using Equation (20):

Embedded Image 24

Thus, under spoofed conditions, the position-domain innovation has the following distribution:

Embedded Image 25

For notational simplicity, we also define the following:

Embedded Image 26

The tracking error affects the position-domain innovation by increasing its variance without changing the mean (which remains zero).

Now, we can write the position-domain innovation before and after spoofing, respectively, as follows:

Embedded Image 27

Embedded Image 28

When normalized by Embedded Image, the position-domain innovation is distributed in the unspoofed case as follows:

Embedded Image 29

In the spoofed case, the position-domain innovation is distributed as follows:

Embedded Image 30

3.4 Sum of Squares of Normalized Position-Domain Innovations

The square of a scalar normal random variable distributed as Embedded Image follows the Gamma distribution Embedded Image (Mathai & Provost, 1992). The Gamma distribution Embedded Image is defined by its shape parameter ρ and scale parameter θ. Therefore, we can write the distribution of the square of the normalized position-domain innovation in the unspoofed case as follows:

Embedded Image 31

In the spoofed case, we obtain the following:

Embedded Image 32

For a period of accumulation N, following the results in Sections 3.2 and 3.3, we define our optimal CPI test statistic (in the unspoofed case) as follows:

Embedded Image 33

We note that the sum of independent Gamma-distributed random variables is also Gamma-distributed (Mathai & Provost, 1992). We assume that the change in variance of the position-domain innovations is negligible over the accumulation period, i.e., Embedded Image and Embedded Image. The test statistic in the unspoofed case qN is then Gamma-distributed as follows:

Embedded Image 34

For a given probability of false alarm PFA, we can determine the threshold as follows:

Embedded Image 35

where Embedded Image is the inverse CDF of the Gamma distribution.

In the spoofed case, with the tracking error embedded in the test statistic, we have the following:

Embedded Image 36

By defining the ratio Embedded Image, we can rewrite the above equation as follows:

Embedded Image 37

From Equations (34) and (37), we can see that tracking error causes the scale parameter of the test statistic distribution to change but does not affect the shape parameter, which remains the same as in the unspoofed distribution. The probability of missed detection (i.e., of not detecting the tracking error) is as follows:

Embedded Image 38

where TN is the threshold, as defined in Equation (35), γ(a, b) is the lower incomplete Gamma function:

Embedded Image 39

and Γ(a) is the Gamma function: Γ(a) = γ(a, ∞).

3.5 Gaussian Approximation

For a Gamma distribution with shape parameter α and scale parameter θ, the mean and variance are αθ and αθ2, respectively. To determine how the tracking error helps with detection, we take the large-N approximation for a Gamma distribution:

Embedded Image 40

Thus, the distributions of Embedded Image and Embedded Image can be approximated for large N, respectively, as follows:

Embedded Image 41

Embedded Image 42

We now introduce a modified test statistic:

Embedded Image 43

The unspoofed distribution of this new test statistic is as follows:

Embedded Image 44

The spoofed test statistic distribution is given as the following:

Embedded Image 45

It is clear from the preceding two equations that the accumulated tracking error causes the mean of the test statistic distribution to grow as N increases while the variance of the distribution remains independent of N.

3.6 Tracking Error as Colored Noise

In the development thus far, for clarity, we have assumed that the tracking error is WGN. However, it is possible that the tracking sensor error could be time-correlated or even that the spoofer would choose to filter the tracking sensor output to try to smooth out the errors to lower the possibility of detection.

To analyze the performance of the CPI monitor, we need to determine the distribution of the test statistic in the case of colored tracking error. We model the tracking error as a zero-mean first-order Gauss–Markov random process (GMRP) with time constant τt and variance Embedded Image. To the spoofer’s advantage, we ignore any time delays that might be incurred in the filtering process.

Although the CPI monitor operates sequentially in time, to analyze the monitor performance, it is more convenient to use a batch method for quadratic forms of random variables, as described by Mathai & Provost (1992). Let X be an N × 1 random vector distributed as Embedded Image and A be an arbitrary N × N symmetric matrix. Then, the quadratic form Q(X) = XTAX can be expressed as a linear combination of independent central chi-squared variables Y:

Embedded Image 46

where λ1, λ2, λ3,…, λN are the eigenvalues of Σ1/2 AΣ1/2, Z = Σ1/2 X, Y = PTZ, and P is the orthonormal matrix of eigenvectors of Σ1/2 AΣ1/2.

We can now express the cumulative position-domain innovation from time 1 to N in terms of X, with Σ defined as follows:

Embedded Image 47

and matrix A defined as follows:

Embedded Image 48

making Q(X) the sum of the squares of the normalized position-domain innovations, which is our CPI test statistic defined in Equation (36). For authentic signals, only the first term in Equation (47) exists because the innovations under normal conditions are white (and Embedded Image). In the case of spoofing, Q(X) includes colored noise, and in light of Equation (46), it will have a generalized chi-squared distribution, whose CDF can be evaluated using the method described by Das & Geisler (2016).

4 QUANTITATIVE RESULTS

We first performed en route simulations of an aircraft utilizing a navigation-grade IMU (Appendix F) and single-frequency Global Positioning System (GPS) measurements. The aircraft level flight was simulated to start from 41°50’10” N, 87°37’30” W with a cruising speed of 454 knots at an altitude of 40,000 ft. The GNSS measurements were generated using the GPS constellation (SC-159, 2020). The error models for the GPS measurements are defined in Appendix A, and the IMUs are listed in Appendix F. The spoofer tracks the aircraft, with errors, to generate and broadcast counterfeit spoofed signals. The simulated spoofed signals are exact replicas of the authentic signals with either additive zero-mean WGN or zero-mean colored noise modeled as a first-order GMRP. In this example, we used a maximum monitor run time of 180 s, a GPS update frequency of 2 Hz (i.e., N ≤ 360), and a false alarm requirement of 10−5. Although tracking errors would exist in all three spatial dimensions, we conservatively assumed that the errors are only in the vertical direction and ignore any errors in the other two directions in the analysis. Figure 1 (left) shows the results of the simulation run with the tracking error modeled as WGN with standard deviation σt = 2 cm. The superior performance of the CPI monitor relative to the CI monitor is plainly evident in this simple example. Figure 1 (right) also illustrates the performance of the CPI monitor for WGN tracking errors with larger magnitudes.

FIGURE 1
FIGURE 1

Performance of the CPI monitor versus the CI monitor for σt = 2 cm (left) and for two different tracking error magnitudes (right)

The results of the direct simulations are instructive; however, such simulations are an unfeasible means for computing missed detection probabilities. Instead, we turn to Equation (38) to quantify the analytical relationship between monitor run time N, WGN tracking error standard deviation σt, and probability of missed detection PMD. The results, for the same en route aircraft scenario, in Figure 2 show that the missed detection probability decreases with increasing run time as more tracking errors are accumulated over time and shift the mean of the test statistic distribution, as predicted in Equation (42). Similarly, an increasing tracking error magnitude (standard deviation) contributes to a shift in the mean of the test statistic, which reduces PMD. Figure 3 shows two-dimensional cuts of Figure 2 for different tracking error variances and monitor run times.

FIGURE 2
FIGURE 2

CPI probability of missed detection PMD versus tracking error σt and monitor run time N

FIGURE 3
FIGURE 3

CPI probability cuts of missed detection PMD versus monitor run time N and tracking error σt

For comparison, we computed the missed detection probabilities for different grades of IMUs. Figure 4 shows the variation in the probability of missed detection as a function of monitor run time N for different IMU grades. The WGN tracking error standard deviation for this example was 10 cm. The figure shows that, even with a relatively low-grade (automotive) IMU, the degradation in monitor performance is negligible.

FIGURE 4
FIGURE 4

Probability of missed detection PMD versus monitor run time (N) for different IMU grades, with σt = 10 cm

The sensitivity of the CPI monitor to tracking error can be attributed to the precision of GPS carrier phase measurements and the IMU’s accelerometer quality, specifically the velocity random walk (VRW). If tracking errors are smaller in magnitude than the relative position errors obtainable from time-differenced carrier phase measurements, the monitor threshold will be too loose to reliably detect a tracking error. Similarly, if the VRW is large, then the position estimate drift prior to the GPS measurement update will be greater than the impact of the tracking error on the measurement itself. Figure 5 illustrates this case by comparing the performance of the CPI monitor in a low-tracking-error scenario, σt = 2 cm, for different IMU grades against a “lousy” IMU with a VRW 100 times greater than that of an automotive-grade IMU. The reason that the automotive-grade IMU performs so well is that the accelerometer noise parameters are comparable to those of higher-grade IMUs. The latter are designed primarily to improve gyroscope bias stability, which is a critical performance parameter for controlling drift over longer time periods—significantly longer than those relevant to the spoofing monitor.

FIGURE 5
FIGURE 5

CPI monitor performance with σt = 2 cm for navigation-, tactical-, and automotive-grade IMUs (left) and automotive versus “lousy” IMUs (right)

We now turn our attention to quantifying the performance of the monitor in the presence of time-correlated (colored) tracking error, coming either from the sensor itself or as a result of filtering of white sensor output by the spoofer. Figure 7 presents example tracking errors over time with the same standard deviation, but different time constants, including WGN. Using the method described in Section 3.6, we obtain the performance results in Figure 6, which show that time-correlated tracking error would help the spoofer remain undetected for longer, with larger filter time constants leading to a higher probability of missed detection.

FIGURE 6
FIGURE 6

Probability of missed detection PMD versus monitor run time N for WGN and colored noise tracking error

The sample interval is 2 Hz; thus, N = 1 corresponds to 0.5 s.

FIGURE 7
FIGURE 7

Illustration of tracking error versus time for white and colored noise processes

We also evaluated the monitor performance for different start times during a standard 24-h period (SC-159, 2020) to evaluate the monitor sensitivity to different satellite geometries. Figure 8 shows the probability of missed detection versus monitor run time for tracking error modeled as WGN with a standard deviation of 10 cm, with 24 curves representing the performance results at 1-h intervals. It is evident that the monitor is effective over the entire day, although there is obviously some variation in performance with satellite geometry.

FIGURE 8
FIGURE 8

Probability of missed detection (PMD) versus monitor run time (N) for different satellite geometries

These results demonstrate that if a CPI monitor is implemented, the conjecture that INS-based spoofing detection is vulnerable to slowly deviating counterfeit signals can be largely dismissed. The results show that the spoofer’s target tracking error should be easily detectable as long as the duration of spoofing lasts exceeds a minimum time defined by the variance and time constant of the tracking error. For the example application considered, Figure 6 shows that even for tracking error standard deviations as small as 10 cm and correlation time constants as large as 40 s, the probability of missed detection is negligible after less than 1 min. We have also experimentally validated the INS monitor, as reported by Kujur et al. (2023), using real GNSS and INS data.

The CPI monitor has the notable limitation that PMD—or equivalently, a position-domain protection level—is difficult, if not impossible, to establish in real time. Thus, the CPI monitor performance must be evaluated in advance by simulation over a wide variety of satellite geometries and tracking error characteristics— for example, by following a more extensive version of the evaluation presented in this paper for the specific GNSS/INS integration and application considered. In the following section, we introduce a complementary SS monitor concept to ensure the detection of faster-onset spoofing, provide the means to produce protection levels in real time, and allow the navigation function to default to INS-only coasting to preserve continuity (at lower accuracy levels as the INS drifts over time).

5 SS MONITOR WITH SEQUENTIAL WINDOWS

The SS monitor measures the difference in position solutions between the potentially spoofed integrated INS/GNSS KF output Embedded Image and an ostensibly clean “coasting” INS-only solution Embedded Image. The test statistic at any time k is defined as follows:

Embedded Image 49

The variance for the test statistic is given by the following:

Embedded Image 50

where Embedded Image and Embedded Image are the estimate error covariance matrices for Embedded Image, respectively. The derivation of Equation (50) is provided in Appendix E. The detection threshold for the test statistic can be obtained given a desired false alert requirement and inverse CDF of Embedded Image. After a detection, if the INS coasting solution is known to be unaffected by earlier spoofing, navigation can continue by switching from Embedded Image to Embedded Image.

Because the SS monitor observes the instantaneous error between the tightly coupled INS/GNSS KF position solution and INS-only coasting solution and because the coasting covariance Embedded Image drifts over time, the detection threshold must continually increase to maintain a constant false alert probability. Thus, as the coasting window length increases, the SS spoofing detection performance will degrade, allowing slowly growing faults to go undetected. While this is obviously not a desirable performance characteristic, it is precisely over these increasing time windows that the CPI monitor will perform the best. Therefore, given the complementary nature of the two monitors, it is natural to consider running them in parallel, where the CPI monitor would detect slowly growing spoofing profiles and the SS would detect more rapidly growing faults.

For the CPI monitor, a minimum window length Nmin is required to achieve a desired Embedded Image. Here, the subscript “+” designates the missed detection (integrity risk) requirement allocation for spoofing attacks lasting longer than Nmin. The CPI monitor threshold would be obtained by using the false alert (continuity risk) allocation for the monitor, Embedded Image. For shorter attack windows, the SS monitor would provide the means for detection with the allocated missed detection and false alert probabilities, Embedded Image and Embedded Image. Obviously, the sum of the two missed detection probabilities must be smaller than the total integrity risk allocated to undetected spoofing. The same is true for the sum of the two false alert probabilities relative to the total continuity risk allocated to spoofing monitoring.

The SS monitor can provide a protection level for spoofing in the spatial direction u as follows:

Embedded Image 51

where:

Embedded Image 52

and the multipliers Embedded Image and Embedded Image are determined from the SS false alert and missed detection requirement probability allocations. The SS detection threshold is Embedded Image. The protection level will increase with the SS window length because of the growth in Embedded Image over time.

The minimum run length Nmin for the CPI monitor sets the upper limit on the SS monitor time window. The combined monitor system is implemented using consecutive fixed-length windows of length Nmin (Figure 9) with an SS monitor initialized at the beginning of each window and terminated at the end of each window. A new SS monitor window is opened at each new GNSS measurement epoch and closed Nmin epochs later. Figure 9 illustrates this idea, showing the (conceptual) protection level. Because each window has its own INS-only solution (from the SS monitor), if spoofing is detected in a particular window at time td, the final INS/GNSS KF solution and associated PL from a prior window closed (without detection) any time before td can be safely used to initialize subsequent fault-free coasting.

FIGURE 9
FIGURE 9

Illustration of SS sequential monitor windows with increasing protection level (PL)

At any given time point, a set number of monitor windows will be running, and the false alert requirement allocation for each monitor can be equally divided among these monitor windows to determine the thresholds for each. This approach is conservative, as the test statistics for the different monitors will be correlated, but it is easy to implement.

The maximum protection level produced by any SS monitor will occur at the end of the window, k = Nmin, because the INS coasting errors, Embedded Image, will be the largest at that point.

The value of Nmin is determined by the CPI monitor performance. For the same example GPS/INS implementation and satellite geometry considered in Figure 6, Figure 10 shows the values of Nmin for different tracking error magnitudes with the missed detection requirement set to 10−7. As in Figure 6, it is clear that colored tracking error takes longer to detect than white noise. Using these values of, we can now determine the maximum protection levels. Figure 11 shows the results for a false alarm requirement of 10−5 and missed detection requirement of 10−6, based on Equation (51). Again, it is clear that the WGN tracking error allows for tighter protection levels compared with colored noise. The large SS protection levels at small values of tracking error σt are due to correspondingly large values of Nmin, which, in turn, lead to large coasting errors Embedded Image (with k = Nmin). For colored noise, a shorter time constant provides a tighter protection level. Moreover, it can be seen that for a decimeter-level tracking error magnitude, the maximum protection level converges regardless of whether the tracking error is time-correlated.

FIGURE 10
FIGURE 10

Minimum detection window (Nmin) versus tracking error magnitude for an example satellite geometry

FIGURE 11
FIGURE 11

Maximum protection level (PL) versus tracking error magnitude for an example satellite geometry

When implemented together, the CPI and SS monitors address all of the limitations of the original CI monitor reported by Tanil et al. (2017), Tanil, Khanafseh, et al. (2018), Tanil, Jimenez, et al. (2018), and Kujur et al. (2019). The CPI monitor performs better than the CI monitor because its test statistic is optimized to detect tracking error. The addition of the SS monitor provides a means for detecting short-duration spoofing and for excluding the spoofed solution after detection. The sequential implementation of the monitors ensures that the performance is independent of the spoofing onset time.

6 CONCLUSIONS

In this work, we introduced an optimal CPI monitor to detect spoofing by accumulating tracking error embedded in the spoofer’s signal. We derived relationships between missed detection probability, tracking error magnitude, and monitor run time. We showed that even with decimeter-level tracking error, the monitor can detect spoofing with a low probability of missed detection in less than 1 min. We evaluated the performance of the CPI monitor for both white and time-correlated (colored) tracking error. To compute protection levels and detect short-duration spoofing, we proposed a complementary SS monitor to be implemented in sequential, overlapping windows to compare the integrated INS/GNSS position solution against an INS-only coasting solution. The INS-only coasting element also provides the capability to maintain positioning continuity after detection, albeit at lower accuracy, as the INS drifts.

HOW TO CITE THIS ARTICLE

Kujur, B., Khanafseh, S., & Pervan, B. (2024). Optimal INS monitor for GNSS spoofer tracking error detection. NAVIGATION, 71(1). https://doi.org/10.33012/navi.629

APPENDIX A: TIGHTLY COUPLED INS/GNSS ARCHITECTURE

Using IMU measurements, an INS provides a navigation state vector, which includes the aircraft position vector r with components x, y, z, velocity vector v with components u, v, w, and attitude ϕ, θ, ψ (Euler angles):

Embedded Image A1

An IMU consists of tri-axis accelerometers and gyroscopes to provide measurements of acceleration and body angular rate. In an INS, the IMU acceleration measurements are integrated once to obtain the velocity and then integrated again to obtain the position, whereas attitude is obtained by integrating angular rate measurements. These measurements have errors (biases and noise), causing the state estimate error to drift over time. In a tightly coupled INS/GNSS architecture, a KF uses raw GNSS code and carrier phase measurements to estimate and correct the error in the drifting INS states to provide an integrated navigation solution.

An individual (scalar) IMU measurement ũ has errors such as time-dependent biases and noise. Therefore, it is modeled as a “true” measurement u*, corrupted with a constant bias bc, a slowly varying time-dependent bias-like component b, and additive WGN ηu, as represented in Equation (A2). The constant bias is usually specified as bias repeatability, and the additive WGN ηu is commonly derived from specifications on the VRW for an accelerometer and angular random walk for a gyroscope:

Embedded Image A2

The time-dependent component of the bias b is modeled as a first-order GMRP with time constant τb and driving WGN vb. The driving WGN is derived from the IMU “bias instability” specifications:

Embedded Image A3

The bias dynamics are included in the process model by augmentation of bias states xbias to the aircraft states. Thus, for three different IMU axes, the bias states for both acceleration and angular rate measurements are shown in Equation (A4). Equations (A1) and (A4) show all of the nominal states that are propagated to obtain the INS navigation solution:

Embedded Image A4

We assume that an en route aircraft utilizes single-frequency GPS measurements without any differential corrections; however, this concept is also applicable to dual-frequency multi-constellation GNSS, terminal, and precision approach scenarios. Equation (A5) shows a simplified GPS measurement model in which the code measurement ρ for each satellite is composed of the true range r, satellite and receiver clock biases dtsv and dtrc, code ionospheric delay Iρ, code tropospheric delay Tρ, code multipath mρ, and receiver code thermal WGN Embedded Image. Similarly, the carrier phase measurement λϕ for each satellite is composed of the true range r, satellite and receiver clock bias dtsv and dtrc, carrier ionospheric delay Iϕ, carrier tropospheric delay Tϕ, carrier phase multipath mϕ, carrier phase cycle integer ambiguity Nϕ, and receiver carrier thermal WGN Embedded Image. The code ionospheric delay Iρ is of the same magnitude as the carrier ionospheric delay Iϕ, and the code tropospheric delay Tρ is of the same magnitude as the carrier tropospheric delay Tϕ:

Embedded Image A5

where c is the speed of light in vacuum and λ is the carrier wavelength.

All GPS errors must be included in the measurement in order to be utilized in the KF. The satellite clock offset cdtsv is corrected based on the clock parameters broadcast in the navigation message. After the satellite clock offset correction has been applied, there are still residual errors caused by satellite clock and orbit ephemeris parameter uncertainty. These residual errors rsv are modeled (Gallon et al., 2020) as a first-order GMRP with a time constant Embedded Image of 5 h subject to driving WGN Embedded Image with a standard deviation of 1.8 m. Equation (A6) represents the first-order GMRP model for satellite clock and ephemeris residual errors:

Embedded Image A6

The receiver clock offset cdtrc is compensated by a constant clock offset drift rate model. The clock offset state rrc is modeled to drift with a constant rate Embedded Image over time, as shown by Equation (A7):

Embedded Image A7

where Embedded Image and Embedded Image are WGN inputs to the clock offset and clock offset drift rate, respectively. The variances of these WGN inputs are obtained using typical Allan variance coefficients of temperature-compensated crystal oscillator timing standards. The white phase noise (h0) and frequency random walk noise (h2) coefficients used are 2 × 10−19 and 2 × 10−20, respectively.

For ionospheric delay, we use the ionospheric correction Tiono from the Klobachaur model, which results in residual errors rt, as modeled by SC-159 (2009), with a standard deviation given by Equation (A8):

Embedded Image A8

where Fpp is the obliquity factor and τvert is calculated given the geomagnetic latitude (SC-159, 2009). Because the ionospheric delay is a slowly changing error, it is modeled as a first-order GMRP with a time constant of 40 h (Appendix B) and driving WGN Embedded Image as follows:

Embedded Image A9

The troposheric delay is corrected with the correction model specified in SC-159 (2009), and the residual errors rt in the zenith direction are modeled as a first-order GMRP with a time constant of 20 h and a standard deviation of 0.09 m (Gallon et al., 2021). The zenith error is converted to a slant ranging error using the elevation-dependent mapping function reported in SC-159 (2009). Equation (A10) shows the first-order GMRP model of the tropospheric residual error rt:

Embedded Image A10

where Embedded Image is the driving WGN for zenith tropospheric residual errors.

Being time-correlated, the multipath is modeled as a first-order GMRP with a time constant τm of 25 s and driving WGN vm (SC-159, 2009):

Embedded Image A11

The standard deviation for code multipath error is 5 m (SC-159, 2009), and for carrier multipath error. we assume the standard deviation to be 0.02 m (Misra & Enge, 2012). The receiver code thermal noise standard deviation is taken to be 0.36 m (SC-159, 2009), and the carrier thermal noise standard deviation is assumed to be 3 mm (Misra & Enge, 2012).

Constant carrier phase cycle integer ambiguities, along with all of the above-mentioned residual error states, are included in the GNSS measurement error states:

Embedded Image A12

where n is the number of satellites in view. The final state vector of the INS/GNSS system is as follows:

Embedded Image A13

APPENDIX B: IONOSPHERIC RESIDUAL ERROR MODEL

FIGURE B1
FIGURE B1

Histogram of rates of total electron content change for a quiet solar maximum day (Walter et al., 2004)

Figure B1 shows a histogram of the ionospheric rate based on 24 h of data from Walter et al. (2004). The data are boundable by a Gaussian distribution with a mean of 3.2 x 10−2 mm/s and a standard deviation of 0.96 mm/s. We model the ionospheric residuals using a first-order GMRP such that the obtained rate matches this Gaussian distribution.

After utilizing Equation (A8) from SC-159 (2009) to obtain the standard deviation for the first-order GMRP, it was determined that the time constant that produces the best approximation of the ionospheric rate distribution in Figure B1 is approximately 40 h.

APPENDIX C: EFFECT OF TRACKING ERROR ON COVARIANCE OF INNOVATIONS

We define the spoofer’s uncertainty in the target’s position as the tracking error νt and use the superscript s for terms that are otherwise related to the spoofer or spoofed signal. At time k, the spoofer estimates the user position with tracking error, given as follows:

Embedded Image C14

where xk, yk, and zk give the true user position. We conservatively assume that there is no latency in the spoofer’s tracking and counterfeit signal generation and neglect additive errors due to thermal noise and multipath in the counterfeit signal transmission. We also concede to the spoofer the ability to replicate the true signal-in-space errors seen by the target—namely, satellite orbit and clock errors and atmospheric effects. We also ignore the contribution of the deliberate deviations added by the spoofer to the counterfeit signal. Relaxing any of these conservative assumptions and concessions would only make the spoofing easier to detect at the target and the analysis less complicated. We ignore these effects in what follows and concern ourselves only with the impacts of tracking error in a single, arbitrary spatial direction.

The spoofer projects the counterfeit state vector Embedded Image into the range domain:

Embedded Image C15

where Embedded Image is the tracking error along the spatial direction corresponding to the unit vector u in the state space. This vector can be re-expressed in terms of the “true” range vector, zk, as follows:

Embedded Image C16

The contribution of the tracking error to the innovations is then given by the following:

Embedded Image C17

Rearranging, we have the following relation:

Embedded Image C18

The covariance of the innovation vector with the presence of tracking error can be written as follows:

Embedded Image C19

APPENDIX D: EFFECT OF TRACKING ERROR ON TIME CORRELATION OF INNOVATIONS

The derivation of the Neyman–Pearson optimal test statistic relies on the assumption that the random variables forming the test statistic—the position-domain innovation in our case—are time-independent. To verify the independence of innovations over time, we generate sample auto-correlation functions (ACFs) and estimate the associated time constants. In the absence of tracking error, independence is guaranteed as long as the nominal error models in the KF are accurate. However, we also need to verify whether the assumption is valid if WGN tracking error is present. Figure D2 shows a sample ACF of carrier phase innovations for the case of WGN tracking error with σt = 10 cm using a navigation-grade IMU. The time constant (measured at the 1/e point) of the sample ACF was 0.21 s, which is less than the sample interval of 0.5 s, thus confirming that the innovations are white.

We expect that a lower-quality IMU would have a greater contribution toward the time correlation of innovations because the Kalman gain would give a higher weight to the GNSS measurements than in the high-quality IMU case. Consequently, the tracking error will affect the post-measurement states Embedded Image to a greater degree as well as the subsequent “predicted” measurement Embedded Image used to generate the innovations. To confirm that the innovations are still white, even for lower-grade IMUs, we evaluate sample ACFs of the carrier phase innovations for navigation-, tactical-, and automotive-grade IMUs. Figure D2 shows the results for the case of WGN tracking error with σt = 10 cm. It is clear from the figure that even for an automotive-grade IMU, the innovations still remain white, with a time constant of 0.24 s.

FIGURE D2
FIGURE D2

Normalized ACF plots of the carrier phase innovations for different IMU grades with WGN tracking error (σt = 10 cm)

APPENDIX E: COVARIANCE OF SS TEST STATISTIC

Let us consider an SS monitor that starts at time epoch 1. The KF time and measurement update equations are as follows:

Embedded Image E20

Embedded Image E21

where, for any time epoch k, Embedded Image and Embedded Image represent the states after the KF time and measurement updates, respectively. Φk is the state transition matrix, Lk is the Kalman gain, zk is the GNSS measurement, and Hk is the observation matrix. The associated estimation equations can be determined as follows:

Embedded Image E22

Embedded Image E23

where, for any time epoch k, ēk and êk represent the state estimate error after the KF time and measurement updates, respectively. I is the identity matrix, wk is the process noise vector, and vk is the measurement noise vector. For the SS monitor, parallel INS-only coasting is initiated with the state propagation equation:

Embedded Image E24

where, for any time epoch k, Embedded Image represents the state after the time update in INS-only coasting. Here, we also assume that the state transition matrix does not differ from that of the KF. The time update error propagation for the first time epoch of INS-only coasting is given as follows:

Embedded Image E25

where, for any time epoch k, Embedded Image represents the state estimate error after the time update during INS-only coasting.

Similarly, for the second time epoch, we can write the KF time and measurement update equations as follows:

Embedded Image E26

Embedded Image E27

We can write the propagation equation for INS-only coasting as follows:

Embedded Image E28

The associated KF error propagation equations are as follows:

Embedded Image E29

Embedded Image E30

For INS-only coasting, the state estimate error propagation equation is given by the following:

Embedded Image E31

We now define the following covariance matrices: Embedded Image, Embedded Image, Embedded Image, and Embedded Image.

The test statistic for the SS monitor at time k is defined as follows:

Embedded Image E32

where:

Embedded Image E23

which has the following covariance matrix:

Embedded Image E34

The covariance for the first time epoch is as follows:

Embedded Image E35

The above equation can be further expanded:

Embedded Image E36

For the second and third terms, we use Equation (E23):

Embedded Image E37

From Equations (E22) and (E25), we know that ē1 and Embedded Image are the same for the first time epoch; hence, we have the following:

Embedded Image E38

Similarly, we have the following relation:

Embedded Image E39

Moreover, from the KF equations, we have the following expression for any time epoch k:

Embedded Image E40

Thus, Equation (E36) takes the following form:

Embedded Image E41

Now, the covariance for the second time epoch is as follows:

Embedded Image E42

As before, this expression can be further expanded:

Embedded Image E43

For the second and third terms, we use Equation (E30):

Embedded Image E44

Substituting Equations (E29) and (E31) into Equation (E44), we obtain the following:

Embedded Image E45

By expanding Equation (E45) and using results from Equations (E37) and (E40), we obtain the following relation:

Embedded Image E46

Substituting Equation (E46) into Equation (E43) gives the following:

Embedded Image E47

Hence, following Equations (E41) and (E47), we can write a general expression for the SS test statistic covariance for any time epoch k:

Embedded Image E48

APPENDIX F: SPECIFICATIONS FOR IMU GRADES

View this table:
TABLE F1

Specifications for Different IMU Grades

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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Optimal INS Monitor for GNSS Spoofer Tracking Error Detection
Birendra Kujur, Samer Khanafseh,, Boris Pervan
NAVIGATION: Journal of the Institute of Navigation Mar 2024, 71 (1) navi.629; DOI: 10.33012/navi.629
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