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### Session A1: Augmentation Services, Integrity, and Authentication 1

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**Frequency Domain Overbounding with Multiple Time Series and PSD Estimators**

*Omar Garcia Crespillo, German Aerospace Center (DLR); Steven Langel, The MITRE Corporation, Mathieu Joerger, Virginia Tech*

Alternate Number 3

In this paper, we address practical aspects of Frequency Domain Overbounding (FDO). First, we develop a high-fidelity modeling method to account for multiple time-series of a stochastic process; multiple time-series may need to be considered because realistic random processes, such as GNSS measurement uncertainty due to satellite orbit and clock ephemeris errors, may only be wide-sense-stationary over a finite time. Second, we quantify the error correlation model sensitivity to power spectral density (PSD) estimation method using theoretical PSD formulas.

Safety-critical navigation applications require that estimation errors be reliably quantified. Over the last two decades, significant effort was spent towards guaranteeing bounds on Global Navigation Satellite Systems (GNSS)-based position estimation errors in the context of satellite-based and ground-based augmentation systems (SBAS, GBAS) and Advanced Receiver Autonomous Integrity Monitoring (ARAIM) for aviation applications. Positioning error bounds were achieved by careful modeling of GNSS ranging measurement errors and by rigorous algorithm design that quantify integrity and continuity risks.

Emerging high-accuracy, high-integrity, and high-continuity navigation applications require the use of sequential estimators such as Kalman filters (KFs). This poses new challenges for integrity monitoring because the time correlation of measurement errors must be robustly accounted for despite being uncertain. Bounding models for uncertain first-order Gauss Markov processes (FOGMP) are derived in the time and frequency domains in Langel et al. (2021); Garcia Crespillo et al. (2023) and are applied in GNSS/INS implementations in Garcia Crespillo et al. (2020); Garcia Crespillo (2022). The more general case of non-FOGMP stationary time-correlated processes is treated in references Langel et al. (2020) and Gallon et al. (2022), which show that PSD upper bounding of independent measurement error sources guarantees an upper bound on KF estimation error variance. Using PSD upper bounding, finite-parameter models such as two-parameter FOGMP models can be used to safely account for complex random processes. Recently, in Joerger et al. (2023) a systematic approach to estimate an empirical PSD from experimental data to ensure high-integrity error modeling was also introduced along with the discussion of practical problems when applying PSD upper bounding. For instance, in Joerger et al. (2023); Langel et al. (2024), for a single time series, an automated method is proposed to automatically find robust autoregressive models up to order two for correlated noise.

There are unresolved practical considerations when applying Frequency Domain Overbounding (FDO) to real discrete sensor data, two of which we address in this paper:

- Multi-signal: he stochastic behaviour of sensor errors may be different in different situations and may also only be stationary over finite time intervals. Thus, if a single model is sought to conservatively represent sensor error in any situation (e.g., to avoid complexity), multiple time series must be considered for model development. Examples of where multiple time series must be considered are the derivation of dynamic models for orbit and clock error of different GNSS satellite constellations or blocks (Gallon et al., 2019), GNSS airborne multipath modeling for different satellite geometries (Crespillo et al., 2024), and the development of inertial sensor error models under varying temperature or vibration conditions

- Multiple PSD estimators: We have previously established that upper bounding in the frequency domain provides overbounding in the state domain for linear sequential estimators like the KF (Langel et al., 2020), provided that the true PSD function of measurement and process noise are known. In practice, it is not possible to obtain the true PSD and instead, one must rely on an estimated PSD. Available methods include the windowed Fourier transform of autocorrelation sequences (Langel et al., 2020) and the use of average periodograms or Welch’s method. Each method will provide its own unique estimate of the PSD that differs from the true PSD. This begs the question as to whether a model that overbounds the estimated PSD is guaranteed to overound the (unknown) true PSD..

The FDO concept was first proposed in (Langel et al., 2020). Since then, this criteria has been applied to model errors in GNSS orbit and clock predictions (Gallon et al., 2022), residual tropospheric delay (Gallon et al., 2021), airborne multipath (Crespillo et al., 2024) and SBAS corrections (Montloin et al., 2023). The current approach for defining overbounding noise models consists of transforming a single time series to the frequency domain by using a PSD estimator and then visually finding a model that upper-bounds the PSD for all frequencies.

In (Langel et al., 2024), an automatic process was developed to find an optimal (variance minimizing) overbounding model (in the form of an autoregressive model up to order two) for a single PSD. While the approach in (Langel et al., 2024) rigorously accounts for the continuous nature of the PSD, it can only be applied (for now) to a single PSD function.

In this work, we develop a methodology to derive a random process model (e.g., FOGMP or autoregressive model of order 1 – we use the notation AR(1)) that overbounds multiple PSDs. The method combines (i) the approach in (Langel et al., 2024) to derive individual AR(1)-bounds for each PSD and (ii) the analytical PSD-upperbound for uncertain FOGMP in (Garcia Crespillo et al., 2023).

First, the automatic FDO of n individual time series produces FOGMP models with different variances and time constants. Second, those parameters define a range of possible values. Their minimum and maximum values can be plugged in the closed form expressions in (Garcia Crespillo et al., 2020) to derive an overall tightly upper-bounding FOGMP model.

Sample-based PSD estimators that are used in practice produce PSD curves that differ from those derived using theoretical continuous-time PSD formulas (Joerger et al., 2023). Time-series Windowing functions and available number of samples impact the shape of the computed PSD.

So far, PSD-upper bounding methods used convenient theoretical continuous-time FOGMP PSD formulas to determine the model whose PSD upper-bounds the experimental curves. The fact that these theoretical formulas change significantly when accounting for sample finiteness (Joerger et al., 2023), questions the validity of the resulting model. In this paper, we evaluate the impact on PSD upper-bounding models of different PSD estimators. We derive new analytical solutions for different window types and PSD estimators that are widely used in signal processing.

This paper provides two key refinements to PSD-upperbounding methods for robust time correlated error modeling. This frequency domain overbounding methodology aims at deriving sensor errors models, which can then be incorporated in sequential estimators to quantify and predict bounds on navigation errors.

REFERENCES

Crespillo, O. G., Gonzalez, R. O., and Caizzone, S. (2024). Airborne time-correlated gnss multipath error modeling of carrier-phase smoothed code. In Proceedings of the 2024 International Technical Meeting of The Institute of Navigation.

Gallon, E., Joerger, M., Perea, S., and Pervan, B. (2019). Error model development for araim exploiting satellite motion. In Proceedings of the 32nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2019).

Gallon, E., Joerger, M., and Pervan, B. (2021). Robust modeling of GNSS tropospheric delay dynamics. IEEE Transactions on Aerospace and Electronic Systems, 57(5):2992–3003.

Gallon, E., Joerger, M., and Pervan, B. (2022). Robust modeling of gnss orbit and clock error dynamics. NAVIGATION: Journal of the Institute of Navigation, 69(4).

Garcia Crespillo, O. (2022). GNSS/INS Kalman Filter Integrity Monitoring with Uncertain Time Correlated Error Processes. PhD thesis, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne.

Garcia Crespillo, O., Joerger, M., and Langel, S. (2020). Overbounding GNSS/INS integration with uncertain GNSS GaussMarkov error parameters. In Position, Navigation and Timing Symposium (PLANS).

Garcia Crespillo, O., Joerger, M., and Langel, S. (2020). Tight bounds for uncertain time-correlated errors with Gauss-Markov structure. arXiv.

Garcia Crespillo, O., Langel, S., and Joerger, M. (2023). Tight bounds for uncertain time-correlated errors with gauss–markov structure in kalman filtering. IEEE Transactions on Aerospace and Electronic Systems, 59(4):4347–4362.

Joerger, M., Jada, S., Langel, S., Garcia Crespillo, O., Gallon, E., and Pervan, B. (2023). Practical considerations in psd upper bounding of experimental data. In Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023).

Langel, S., Garcia Crespillo, O., and Joerger, M. (2020). A new approach for modeling correlated Gaussian errors using frequency domain overbounding. In Position, Navigation and Timing Symposium (PLANS).

Langel, S., Garcia Crespillo, O., and Joerger, M. (2024). Frequency-domain modeling of correlated gaussian noise in kalman filtering. unpublished.

Langel, S., Garcia Crespillo, O., and Joerger, M. (2021). Overbounding the effect of uncertain Gauss-Markov noise in Kalman filtering. NAVIGATION, 68(2):259–276.

Montloin, L., Legrand, F., Bauer, F., Buscarlet, G., Dall’Orso, M., and de Echazarreta, C. L. (2023). Sbas time-correlated error characterization for sequential position error overbounding. In Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023).

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