The company positions the QLX3Gx as a market-ready receiver built around its Dragonfly Digital RF architecture, which moves many traditionally analog RF functions into the digital domain to reduce power, size and cost while retaining multi-constellation GNSS performance. The new evaluation kit is intended to let OEMs characterize power consumption and positioning behavior in their own devices before committing to volume designs. 

According to Qualinx, the QLX3Gx can operate in a 1 mW GNSS mode, with the same silicon supporting a range of power-versus-performance configurations through software. The chip is designed to track multiple constellations and bands concurrently, and to keep tracking and navigation computation on the chip rather than offloading to cloud services or host processors. It also supports authenticated Galileo signals via OSNMA to improve resilience against spoofing, with the company highlighting use cases in asset tracking, wearables and other edge devices that need long battery life as well as resistance to malicious interference. 

Qualinx is also emphasizing supply-chain and manufacturing aspects, noting that the GNSS chip is fabricated at GlobalFoundries’ facility in Dresden, Germany, as part of a broader European semiconductor footprint. A recent €20 million funding round is intended to help move the QLX3Gx family into volume production and expand its availability in international markets. 

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Built on the company’s X20 high-precision platform, the module delivers RTK-grade performance while maintaining precise GNSS-based heading even at low speeds or standstill, a key requirement for auto-steering and autonomous operation. Target sectors include precision agriculture, unmanned aerial vehicles, autonomous machinery, marine and robotics navigation. 

All-band on both antennas, with scalable corrections

The ZED-X20D tracks all major GNSS constellations on L1, L2, L5 and L6, and adds L-band reception for PPP correction services, an “all band on both antennas” approach that is intended to maximize heading availability and stability in challenging environments. To meet different accuracy and deployment needs, it works with RTK, PPP-RTK and PPP correction services, including u-blox’s PointPerfect offerings for regional and global coverage. Built-in support for Galileo E6 enables use of the free Galileo High Accuracy Service (HAS), giving equipment makers multiple options to source corrections. 

u-blox is positioning the ZED-X20D as a drop-in upgrade for existing designs by retaining the established ZED form factor and pairing the module with its ANN-MB2 all-band antenna and PointPerfect services as a turnkey high-precision bundle. The company says this combination is aimed at simplifying design, reducing system cost and accelerating mass adoption of automated and autonomous equipment across agriculture, UAVs, construction and other industrial domains. 

Security and interference resilience for trusted heading

The module includes u-blox’s end-to-end hardened security, with secure boot, signed firmware and a hardware root of trust for cryptographic material, as well as support for Galileo OSNMA and encrypted correction data. All-band frequency diversity and interference monitoring are designed to improve resilience against jamming and other RF threats, while access to high-quality GNSS measurements supports reliable post-processing and integrity monitoring—features likely to appeal to developers building safety-critical or highly automated systems on top of the new heading platform.

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The new solution, SimXTRACT, allows engineers to decompose real-world RF recordings into discrete signal components and replay them as fully controllable scenarios on Spirent simulators.

Positioning, navigation and timing (PNT) developers have traditionally been forced to choose between RF record-and-playback on one side and pure lab simulation on the other. Record-and-playback captures all the richness of the real world, but offers limited control and repeatability. Simulation provides precise control over parameters and repeatability for regression and corner-case testing, but can lack the full complexity of live-sky environments. Spirent positions SimXTRACT as a way to fuse these two approaches.

According to the company, SimXTRACT takes signals captured in the field using Spirent record-and-playback devices and performs complex signal decomposition, breaking each received signal into separate line-of-sight and multipath ray paths. Metadata such as Doppler offset, code error, power level, and angle of arrival (AoA) is retained. That decomposed representation is then converted into simulator drive files that can be loaded into Spirent GNSS simulators as fully controllable, repeatable scenarios.

“SimXTRACT revolutionizes how you can test positioning solutions. By combining real-world insights with lab-based control and repeatability, our customers will no longer have to compromise on how they test in this fast-moving technology area,” said Peter Terry-Brown, Divisional CEO of Spirent’s Positioning business, in the announcement. “SimXTRACT ensures customers get the best of both worlds, with enhanced realism delivering more accurate results, quicker issue resolution, and faster time to market.”

By reducing the amount of time and number of trips required for field data collection, Spirent says users can cut the cost and complexity of scenario capture and generation while still working with high-fidelity, real-world conditions. Developers can also analyze signal recordings, search for specific types of events or environments, and then recreate those conditions in the lab to focus troubleshooting or performance characterization.

Spirent expects SimXTRACT to be used across a broad set of sectors where high-precision PNT is critical, including automotive, chipsets, consumer devices, defense and critical infrastructure. Terry-Brown frames the tool as a way to “bring the real-world environment into every stage of your product realization process,” with the goal of improving product quality while saving time and money.

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DARPA has outlined new details of its Robust Optical Clock Network (ROCkN) program, describing how high-precision optical clocks could allow U.S. forces to retain GPS-grade positioning, navigation and timing (PNT) in contested environments while also enabling timing precision well beyond what space-based GNSS currently provides. The agency’s latest update, released March 2, emphasizes GPS-free operations for platforms and networks operating under jamming, spoofing or prolonged signal outages. 

Modern missiles, sensors, aircraft and artillery all depend on the nanosecond-accurate timing disseminated by GPS satellites. A timing error of just a few billionths of a second translates into position errors on the order of a meter or more, which can quickly degrade weapon accuracy and sensor coherence. Despite hardening measures, the space-to-ground link remains vulnerable to electronic attack and interference, making assured timing one of the central challenges for GNSS-dependent forces. 

ROCkN Pushes Optical Clocks From Physics Labs Into Tactical Form Factors

ROCkN’s answer is to push optical-domain timing out of the lab and into tactical-grade hardware. The program is developing two classes of clocks: a compact, shoebox-sized unit with power consumption comparable to a household lightbulb, designed to hold GPS-level timing (sub-nanosecond precision) for up to two weeks without any satellite updates; and a larger, “washing-machine”-sized master clock intended to provide a regional time reference with GPS-level precision for more than six months without resynchronization. Both clocks are being engineered under strict size, weight and power constraints for deployment on mobile or forward-based platforms. 

At the network level, ROCkN is also demonstrating over-the-air optical time-transfer techniques that push beyond GPS’ few-nanosecond accuracy. According to DARPA, recent field tests have achieved femtosecond-level synchronization over hundreds of kilometers and have operated multi-node clock networks in challenging weather, from tropical humidity to heat waves and blizzards. Moving from nanosecond to picosecond-and-better timing is expected to unlock new capabilities in distributed sensing, coherent radar and electronic warfare, as well as wideband, high-capacity communications.

Femtosecond-Level Synchronization Targets Next-Gen Sensing, EW and Comms 

The most immediate impact is in timing resilience. Optical clocks with weeks- to months-long holdover effectively create local and regional “mini-time scales” that can maintain GPS-equivalent performance through prolonged outages or deliberate interference. In practice, that means platforms and sensor networks could retain precise time tags and navigation solutions even when denied access to GPS, and could re-align with space-based systems once signals become available again. DARPA also highlights the potential for coherent synthesis of data from multiple compact, mobile sensors at frequencies beyond X-band, improving emitter geolocation and target characterization in contested spectrum. 

DARPA reports that ROCkN hardware has already been flown on fixed-wing aircraft, integrated on ground vehicles and deployed at sea for a three-week demonstration aboard a naval vessel operating in the Pacific. Over the coming year, the agency plans a series of field exercises showcasing ROCkN-enabled capabilities across next-generation PNT, electronic warfare and ISR mission sets, alongside a pilot-line manufacturing effort aimed at supplying Department of Defense transition partners. 

Taken together, ROCkN positions optical-clock-based timing as a key pillar in the broader push toward alternative and complementary PNT. For GNSS users and system designers, it underscores a strategic shift: from relying solely on space-borne signals for timing, to a hybrid model in which resilient, GPS-independent local time sources and optical time-transfer networks backstop – and ultimately extend – the performance envelope of satellite navigation.

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SÉBASTIEN TRILLES, ODILE MALIET, JULIE ANTIC, KIN MIMOUNI, THALES ALENIA SPACE, FRANCE

In a broad sense, the notion of integrity refers to the level of confidence one can have in data obtained from a calculation result. In positioning systems, integrity is a measure of the trustworthiness a user can place in a position estimate. Geolocation would be perfect if measurements were error-free, but this is never the case as all device measurements inherently contain errors and noise. Hence, a discrepancy between the calculated position and the true (but unknown) one always exists. As errors and noise contains stochastic part, integrity is fundamentally grounded in probabilistic theory.

Mathematically expressed, integrity is equivalent to assigning a probability of the estimate being outside a defined confidence interval (protection level). The integrity of a positioning system is compromised when anomalies occur, leading to unexpected positioning errors beyond the operational protection level. These anomalies could persist for more than a few seconds within a specific time interval (T). In such cases, the integrity risk (IR) is defined as the probability the true position remains outside the protection level for a duration exceeding T.

For instance, in precision approach operations in aviation, the Standards And Recommended Practices (SARPs) set the IR at 2×10^-7 per approach (150s) and the specific time interval to T=6s. The computation of Protection Level (PL) consists of scaling position error variance to the integrity requirement using K-factors. The K-factors are derived from statistical laws and are critical for ensuring the system’s integrity in various operational conditions, taking into account the errors’ time-correlation [9] [10].

In practical applications, positioning is accomplished using measurements with well-known residual error structure and statistical distributions. Position and time errors are determined by linearly combining the residual errors of the measurements. The process starts with the GNSS navigation solution involving the estimation of a position-time correction x as a solution to the measurement equations linearized around a given position-time priori: WGx=Wb+ε, where G is a m×4 matrix with m the number of line of sight, ε is the measurement noises vector, W is a weight matrix and b is the residual measurements vector. The noises ε are assumed to follow a Normal centered law.

The so called design matrix G is defined as the matrix of partial derivatives of the measurement equations with respect to the parameters of position and time. The partial derivative of the pseudo range with respect to the position correction is obtained from the partial derivatives of the geometric distance D=||X^s-X_r || between satellite position X^s and (unknown) receiver position X_r. Re-writing the geometric distance as D=u_r^s∙(X^s-X_r), the partial derivative of with respect to the positions is the unit vector of the line of sight–u_r^s from receiver to satellite.

The weights per line of sight are built using different model variances related to various contributor to measurement errors: residual system errors (orbit and clock), propagation errors (ionosphere and troposphere), and local errors (multipath, thermal noise and interference). The maximum likelihood method provides the estimate =Sb, where S=(G^t W G)^-1G^t W is the 4×m sensitivity matrix expressed in the east (E), north (N), up (U) local frame and time.

The estimation positioning error e is given by 

The first three row components of S, respectively s_E,i, s_N,i and s_U,i correspond to the partial derivatives of position errors with respect to the east, north and up directions in relation to the measurement errors of the i-th satellite. The sensitivity matrix linearly projects the unmodeled residual measurement errors ε_i in a given direction.

For east direction:

for north:

and for up:

This process establishes a straightforward mathematical transfer that enables the projection of residual errors from the measurement domain to the position domain. Focusing on measurement domain, we seek sufficient properties regarding measurement errors distribution that ensure integrity in the domain of positions. These properties define integrity at the measurement level. If these properties are respected, integrity in the domain of positions is ensured, referred to as the transfer of integrity from the measurement domain to the position domain.

This article provides a comprehensive overview of the various concepts related to CDF overbounding. It aims to articulate each concept within a common framework, delineate their ranges and limitations, and ultimately present a methodology for creating protection volumes that reliably contain positioning errors with a high level of confidence.

MOPS Integrity Concept 

Initially, integrity concept was developed for aeronautical users and standardized by the Minimum Operational Performance Standards (MOPS) document [1] and is defined at position level. This standard deals with integrity parameters broadcasted by SBAS regarding ionosphere, orbit and clock corrections. In this aspect, MOPS considers the individual measurement error contributors as independent, making it possible to sum up all model variances in unique one σ_i² per line of sight. The weight matrix is defined as diagonal W=diag(w₁,…,w_m) where w_i=1⁄σ_i²).

Therefore, the measurement errors are assumed of white noise type, so their distributions have a zero expectation E[ε]=0, which implies the expectation of the identification error is also zero: E[e]=SE[ε]=0.

The covariance of the error is: cov(e)=E[(e-E[e]) (e-E[e])^t]=E[ee^t]SE[εε^t] S^t=Scov(ε)S^t. Therefore, the minimum covariance is reached by taking cov(ε)=W^-1, thus cov(e)=SW^-1S^t=(G^t WG)^-1. This refers to a four-dimensional symmetric positive-definite matrix whose are linear combinations of the measurement variances, as for instance:

The positioning error structure is then separated into horizontal errors e_H and vertical errors e_U, which amounts to considering the following extracted submatrices:

The MOPS specify the integrity risk IR as the maximum allowable probability for the navigation position error to exceed the alarm limit without the system alerting the user within the alert time. In the case of a standardized Normal distribution, such as N(0,1), the K-factor depends on the risk IR:

By making the change of variables t=√2u in Equation 2, we obtain an expression that depends on the complementary error function erfc:

and after inversion in Equation 3, we get the expression of the usual Gaussian K-factor:

Application to protection volumes: The MOPS standard define the vertical protection volume as:

Where the K-factor inflates the standard deviation d_U at a level compatible with integrity requirements. Because a linear combination of Gaussian-distributed vector is Gaussian-distributed, the residual position errors follow a Normal law. If Φ_eU denotes the cumulative distribution function (CDF) of this Normal law, defined by Φ_eU(x)=P(e_U≤x), we then have:

Equation 6 indicates that the absolute value of the error e_U is bounded by the confidence interval VPL defined in Equation 5 at the probability (1-IR).

The integrity MOPS concept does not mention any overbounding approach. It does not provide information regarding the shape of empirical residual errors distribution. It only mentions [1] the necessity from SBAS to broadcast two parameters, the first one being the variance of Normal distributions associated with the user differential range error for a satellite after application of corrections, and the second one associated with residual ionosphere vertical error at an ionospheric grid point for an L1 signal. The term “associated” as used by MOPS leaves room for several possible interpretations.

In fact, these definitions may give the impression that these Normal distributions represent the actual errors distribution. This interpretation has already been mentioned by several authors [5]. As a consequence, based on the stability of independent Normal distributions through linear combination, the position errors distribution is also represented by a Normal distribution. Unfortunately, the actual range errors are generally not Normal, especially in the tails.

In that context, it could be tempting to check integrity at the pseudorange level by making sure that, for all lines of sight i, the Normal distribution with standard deviation σ_i is conservative at the quantile equals to the integrity risk IR:

Non-intuitively, this naïve approach is not correct in general conditions. Annex A3 in [1] shows a toy example that satisfies Equation 7 for all lines of sight, and yet is not compliant with the integrity risk IR on position. This example clearly shows integrity transfer from range to position is not obvious and explains the emergence of overbounding concepts. 

CDF-Overbounding 

The CDF-overbounding concept has been introduced by [2] in the field of aeronautical users. The main idea is to overbound the empirical measurement error distribution, in the field of CDF, by a simpler one allowing to better control the integrity risk mainly in the tails, in the absence of faults. An overbound can be viewed as a statistic distribution that is a regular envelope of the empirical distribution. It is interesting to note the mathematical results presented in [2] were already known to the statistic community [14].

In the following, the CDF of a random variable X is denoted by F_X. According to [2], the random variable O_X is a CDF-overbound of the random variable X, and we note XO_X, if

The binary relationship (8) defines a partial order on the set of distributions. Indeed, the CDF-overbound relation is:

• Reflexive: XX
(every element is related to itself)

• Transitive: if XY and YZ then XZ
(the order is maintained through the chain)

• Antisymmetric: if XY and
YX then X=Y (two elements can’t mutually precede each other; they are considered equal)

Three graphical representations of CDF-overbounding are provided in Figure 1 for a Gaussian overbound with standard deviation equals to 0.7. The green area represents the domain for the CDF (respectively folded CDF and QQ plot) of X, that satisfies the CDF-overbounding of X by O_X. On the left, the CDF of the overbound O_X is represented in black. This representation is a direct illustration of the definition. The representation in the middle is based on the folded CDF that equals the CDF before the median and the survival function (1-CDF) after the median. This representation is handy to inspect the overbounding on the left and right tails thanks to the log-scale. The representation on the left is based on the Quantile-Quantile (QQ) plot for X and O_X. It permits inspecting both the core and the tails of the distribution.

Introducing the overbounding concept allows the following result: 

Theorem 1: If X and Y are two centered symmetric and unimodal distributions, and O_X and O_Y their respective overbounds are also symmetric and unimodal. Then for all α, β in , the linear combination αX+βY is CDF-overbounded by αO_X+βO_Y. In short, the overbounding property is stable by linear combination:

If the distribution of each residual measurement error is symmetric, unimodal and can be overbounded by a distribution that is also symmetric and unimodal, then the positioning errors are also overbounded by a known symmetric and unimodal distribution. Under this assumption, integrity in the pseudorange domain implies integrity in the position domain.

Proof:

Proof of stability by addition:

The proof is established in two steps. The first step is to prove that if 

The second step is a direct application of the previous statement: if

The sum of two independent symmetric and unimodal variables is itself symmetric and unimodal. The proof is detailed in the Appendix (see online version). This shows all the considered distributions have the right properties to be compared by CDF-overbounding.

Applying the definition of CDF-overbounding, proving (1) comes down to establishing that:

Let us fix z and compute F_X+OY(z)–F_X+Y(z) using the formula for the CDF of the sum of two independent random variables (the derivation of the formula is recalled in the Appendix):

We now split the integral of Equation 10 into two parts and make the change of variable x to -x for the negative part:

Recalling that by requirement of the CDF-overbounding both Y and O_Y distributions are symmetrical, which means F_Y(-x)=1-F_Y(x) and thus F_OY(-x)-F_Y(-x)=F_Y(x)-F_OY(x). Equation 11 becomes:

By definition of 

the right parenthesis of the integrand in Equation 12 is positive for all x. Because X is unimodal and symmetric by assumption, its PDF f_X peaks at f_X(0) and is decreasing on the positive and negative sides. Thus, the unimodality and symmetry implies that, for all x₁ and x₂, if |x₁|≤|x₂| then f_X(x₁)≥f_X(x₂). 

In our case of interest, if z is negative, we have for all positive x, |z+x|≤|z-x| and so f_X(z+x)≥ f_X(z-x), which means the left parenthesis of the previous integrand is also positive. We can deduce that

which is precisely the statement (A). On the other hand, if z is positive, |z+x|≥ |z-x| and so f_X(z+x)≤f_X(z-x) meaning that the integrand is negative and so

This proves (B) completing the proof that, if 

and the distributions X, Y, O_Y are symmetric unimodal, then

By applying the same reasoning, we get that if

and the distributions X, O_X, O_Y are symmetric unimodal, then

This finishes the proof of stability of CDF-overbounding by addition.

Proof of stability by multiplication by a scalar: for all real α,

Let X, O_X be two symmetric, unimodal random variables such that

First of all, αX and αO_X are also symmetric and unimodal. By symmetry of X and O_X, we can restrict ourselves to the case where α is strictly positive. According to the equality F_αX(x)=F_X(x⁄α) and similarly for O_X so the inequalities defining XO_X directly translate to αXαO_X.

Application to protection volumes: If X is the distribution of the residual error (symmetric and unimodal by assumption), then a CDF-overbound O_X allows us to put a lower bound on the probability of the error to be in a certain interval containing 0. More specifically, for a negative a and positive b, we have: . 

Usually, protection volumes are chosen to be symmetric, and thus for positive a,

and so any protection volume computed with the overbounding distribution is a conservative protection volume for the original distribution.

In practice, O_X is a Gaussian distribution, chosen for its stability by linear combinations. If in the total error

each error component ε_i is CDF-overbounded by a Gaussian distribution with standard deviation σ_i, then the total error e is CDF-overbounded by a Gaussian of standard deviation

Thus, the usual formula for the protection . Thus, the usual formula for the protection level can be used, with

Case of distributions with bias: The CDF-overbound theorem requires the distributions of error components to be centered. However, the absence of bias in residual measurement errors is never perfectly satisfied because of systematic errors due to troposphere, multipath inter-channel bias, etc.

These errors along the lines of sight are therefore composed of a random part, a noise ε_i with zero mean, plus an additional bias μ_i. If these biases were known, they would be integrated into the SBAS corrections, but this is not the case. However, we will assume we know a bound on their absolute values.

The absolute value of the position error along one coordinate is then given by:

We can build the protection volume that covers the suffered errors as follows:

where the factor K is computed according to the error distribution. In the Gaussian case, the multiplicative factor K is calculated using the inverse of the complementary function K=√2erfc^-1(IR).

The question of introducing biases as a multiplicative factor of the protection volume calculated with the classic formulation arises. To do this, we introduce the following calculation:

Here, the quantities

We then find the desired multiplicative factor:

This multiplicative factor ξ makes it possible to inflate the classic volumes of protections (expressed without bias) so as to encompass these residual biases:

This approach allows us to state the following result:

Theorem 1bis: If the distribution of each of the residual measurement errors is symmetric around its median, unimodal and can be overbounded by a Gaussian distribution with the same median (i.e. the mean of the Gaussian is equal to the median of the residual error distribution), then the protection level for the position error can be computed with the usual formula, provided we inflate the K-factor by the multiplicative factor ξ. Under these assumptions, integrity in the pseudorange domain imply integrity in the position domain.

The definition of coverage relative to a median is given by:

The concept of CDF-overbound requires the tails of the overbound cover the tails of the empirical distribution. In the case of a Gaussian overbound, we know the tails of the Gaussian distributions are very light and we will end up finding a quantile (even if it is very large) beyond which the tail of the Gaussian passes below the tail of the empirical distribution. This is why we set in practice a quantile q, beyond the specified integrity risk q>F^-1(IR⁄2N), within which, on the interval [-q,q], the overbound property is verified.

Paired Overbounding 

This concept was introduced in [3,4] to relax the strong assumptions of CDF-overbounding, namely that the distributions of the residual error components are centered, unimodal and symmetric.

Definition: The random variables L_X and R_X are a paired overbound of the random variable X, and we note X ⊆ [L_X, R_X] if 

The random variable L_X is said to be the left overbound and R_X is said to be the right overbound of the random variable X. Figure 2 displays three representations of paired overbounding by two Gaussian distributions with standard deviation 0.7 and bias equals to -/+ 0.3, adopting the same conventions as in Figure 1. 

Theorem 2a: For independent random variables X,Y, if X ⊆ [L_X,R_X] and Y ⊆ [L_Y,R_Y] then X+Y ⊆ [L_X+L_Y,R_X+R_Y]. In other words, the pair overbounding concept is stable by convolution. A priori the random variables X,Y and their overbounding pair are arbitrary; no particular assumption is necessary to demonstrate stability by convolution, which is the strength of the concept.

Proof:

We have, using the definition of the left-overbounding:

And so ∀z ε,F_X+Y(z)≤F_X+LY(z). Repeating the same argument:

Combining the two inequalities of Equations 25 and 26, we get: 

The proof of the inverse inequality is established in a similar way without difficulties. 

Theorem 2b: If X⊆[L_X,R_X], then for a positive real α we have αX⊆[αL_X,αR_X]. However, for a negative real α, we have αX⊆[αR_X,αL_X]. If we further require that the over-bounding pair is symmetric, meaning that L_X=-R_X, we can write the result as follows: for all α, if X⊆[-R_X,R_X], then αX⊆[–|α|R_X,|α|R_X].

Proof:

We have for positive α:

On the other hand, for a negative α, we have:

These theorems show the stability of linear combinations is satisfied with any pair of overbounds (possibility asymmetric or multi-modal) but only with positive coefficients. Unfortunately, this property is not sufficient to guarantee the integrity transfer from range to positioning domain (because the geometry coefficients are signed). The following Theorem permits to guarantee the stability of any linear combinations (with positive or negative coefficients) by adding a condition on the pair of overbounds that is L_X=–R_X.

Theorem 2c: If X⊆[–R_X,R_X] and Y⊆
[–R_Y,R_Y] then ∀(α,β)∈², αX+βY⊆
[–|α|R_X–|β|R_Y,|α|R_X+|β|R_Y]. If the distribution of each of the residual measurement errors is paired-overbounded by a symmetric pair, then the position errors are also paired-overbounded by a known pair. Under these conditions, integrity in the pseudorange domain imply integrity in the position domain.

This approach is useful for taking residual biases into acount. Consider a given line of sight and a contributor X to the residual errors on this line of sight. This contributor presents a residual bias for which a known bound is μ.

We construct a bounding of distribution X by two Normal laws with standard deviation σ, the left bound biased by -μ and the right bound biased by +μ, which form a symmetric pair:

If we collect all the lines of sight and each contributor to the measurement errors paired-overbounded, then the convolution property implies that the position error is also a paired-overbounded distribution with a variance and a bias given respectively by

The limiting aspect is the search for left and right bounds can bring conservatism in practice, knowing these boundaries must frame the entire empirical distribution.

Application to protection volumes: The pair overbounding X⊆[L_X,R_X] allows us to put a lower bound on the probability to be in a certain interval: for any a,b with a<b, we have:

In the case of an overbounding pair by two Gaussians of mean ±μ and variance σ², we have for a symmetric protection level:

and so the integrity condition P(X∈
[-PL,PL])≥1-IR can be ensured for

If in the total error 

 each error component ε_i is pair-overbounded by two Gaussian distribution with standard deviation σ_i and mean ±μ, then by the stability by linear combinations, the total error e is pair-overbounded by two Gaussians of standard deviation σ= and mean μ=. Thus, the formula for the protection level is identical to the formula in the previous section, but with different hypothesis on the original distributions. 

Core-Tail Overbounding 

In general, the empirical distribution of interest is obtained by collecting large volumes of data. This is sufficient to accurately represent the core of the distribution but there is always a point where the tail remains unknown because the collected samples are always finite. Therefore, how can we ensure the constructed overbounding distribution remains correct for the entire underlying distribution?

Furthermore, if the tail of the underlying distribution is known analytically or with good numerical precision, the overbounding theorems presented impose a condition on the entire distribution. When using Gaussian overbounds, which have very light tails, this can lead to excessive conservatism, for example to absorb some mass far in the tail, or can be mathematically impossible if the analytic expression of the underlying distribution falls slower than a Gaussian.

The strategy presented by [6] is a mean to deal with these two problems. It divides the cumulative distribution function F_OX of the overbound distribution into two parts: an explicit core and an implicit tail. The overbounding distribution is a mixture of both, and the value this function takes at a point is equal to the sum of the core cumulative distribution function F_OX,c and the tail cumulative distribution function F_OX,t, weighted by the probability P_t that the point is in the core or the tail:

The core distribution is explicit, meaning it can be expressed analytically and calculations performed. Most often, it is a Gaussian distribution. As for the tail, which poses the most problems, it is left completely arbitrary: We know it exists, but we do not wish to perform calculations with it. Instead, we will always consider the worst possible tail for the chosen application to constrain the maximum impact of the unknown tail.

This approach is valid for both CDF-overbounding and paired overbounding. The benefit of this decomposition is to be able to focus on the core overbound, while ensuring the unknown tail of X is defined such that F_OX(x) (defined by Equation 34) is an overbound of X, with the following approach of F_OX,t as a pseudo CDF:

Intuitively, inequality (24) means the worst tail for the CDF-overbounding is a probability weight of 1⁄2 localized at both infinities, whereas the worst tail for a left-overbound is to have all probability concentrated at minus infinity (plus infinity for the right overbound). Formally, the resulting overbounding function F_OX(x) is not a CDF (because its total weight is not 1) but as in the excess mass concept, all properties of the corresponding overbounding function remain unchanged.

In practice, a CDF-overbound in the core-tail overbounding concept consists of a (symmetric unimodal) function F_OX,c and a tail weight P_t such that F_OX=(1-P_t) F_OX,c+P_t⁄2 (because for CDF-overbounding the tail CDF is always chosen as a constant of value 1⁄2) is a CDF-overbound of the distribution of interest X. In other words, we need F_OX,c and P_t such XO_X. For pair-overbounding, the left and right bounds are split into core and tail. We need a left and right core overbounding distribution F_LX,c and F_RX,c and a tail weight P_t such that:

Graphical representations of core-tail overbounding is illustrated with P_t equals to 5×10^-2 in Figure 3 for CDF-overbounding by a Gaussian with standard deviation 0.7, and Figure 4 for paired overbounding by two Gaussian with biais +/- 0.3 and standard deviation 0.7. The green area represents the domain that satisfies the overbounding of X by O_X.

Note the core-tail overbounding is a weaker condition than the original overbounding condition, in the sense that any overbounding distribution (CDF or paired) can be seen as the core overbounding distribution with a tail weight P_t, whatever the value of P_t. The downside is the resulting protection volumes will be more conservative for larger tail weight, and can be undefined if the integrity risk is lower than the tail weight. This phenomenon allows for the following result as part of this new paradigm.

Theorem 4: Let X and Y be two random variables that admits a CDF-overbound (central or paired) that can be decomposed into a core part and a tail part with weight P_t,X and P_t,Y. Then the linear combination αX+βY admits a corresponding overbound with the core given by the theorem 1 or 2c, and with core weight (1-P_t,X)×(1-P_t,Y). Under these conditions, integrity in the pseudorange domain imply integrity in the position domain.

Proof:

We will first consider the framework of the central CDF-overbound and the overbounding of X+Y. Theorem 1 (stability by convolution) gives us X+Y is overbounded by O_X+O_Y (the core-tail overbound remains a centered symmetric and unimodal distributions):

We then inject into inequality (26) the decomposition of the overbounds of the random variables X and Y into core and tail.

By expanding expression (38), we get

On the other hand, the cumulative distribution function of the overbound of the sum is decomposed into core and tail parts:

By identification with Equations 38 and 39 we find that:

1-P_O,t=(1-P_t,X)(1-P_t,Y), which gives the weight of the core,

F_OX+O_Y,c=F_OX,c*f_OY,c, so the core of the sum is the convolution of the cores of each distribution,

The rest of the expression is considered as the tail and its explicit expression is not needed.

The last step is to prove by replacing the implicit tail part by the constant 1⁄2, we still have a CDF-overbounding of the sum. For x≤0, the three terms F_OX,t*f_OY,c(x), F_OX,c*f_OY,t (x), F_OX,t*f_OY,t (x) are smaller than 1⁄2 because each one is a symmetric CDF. Thus, for negative x, F_OX*f_OY(x)≤(1-P_O,t)×F_OX+O_Y,c+P_O,t⁄2. On the positive side, the equations are reversed, and we get a CDF-overbound of the sum by considering only the core part of the two distributions. We can replace the tail part by its generic value.

The multiplication by a scalar is treated as in Theorem 1 and does not change the weight of the core or tail. 

In the pair overbounding case, the resulting pair is symmetric if the core overbounding pair is symmetric, so formula of theorem 2c holds as long as each individual overbounding pair has the same core and tail weights for the left and right overbounding pair. The proof is very similar to the CDF-overbounding case.

The core-tail overbounding concepts allows us to manipulate a weaker form of CDF or paired-overbounding. It is weaker because the inequalities are not required on the entire CDF but only on the core part. The properties of stability by linear combination of the CDF and paired-overbounding are maintained but at the price of a small “contamination” of the tail for each added term. For each addition, the tail weight grows. An upper bound on the weight of the tail can easily be derived as follows: In the case where all weights are equal with tail weight P_t, we have P_O,t=1-(1-P_t )²<2P_t, and by recurrence on n sources of errors, we have P_O,t<nP_t.

Application to protection volumes: The protection volume formulas are identical to the CDF or paired-overbounding cases, with the replacement of the overbounding distribution by the core plus tail part. If we are working with Gaussian distributions, the properties of stability by linear combination allows us to use the same formulas by changing only the value of the K-factor, which now depends on the weight of the tail distribution. The compatible quantile K of a given integrity risk IR in the position domain is calculated by considering the CDF of the overbounding function and inverting it. The Gaussian K-factor changes for the CDF-overbound to:

For a protection volume to be calculable with the K-factor, it is necessary that IR-P_O,t>0 and thus that P_O,t<IR. Otherwise, the knowledge of the core distribution is not sufficient to build a suitable protection volume and the protection level goes to infinity, and gives no information in practice.

For the paired CDF-overbound the K-factor changes to:

Note the weight of the tail depends on the number of error contributors. If each error has an overbound tail weight P_t<<1, then the full error overbound has tail weight nP_t. This means the density of the tail distribution of each measurement error must be at least n times lower than the selected integrity risk for the position domain. In case this condition is not met, the protection volume is undefined (infinite) because the knowledge of the core distribution is not sufficient to guarantee a protection level at the given integrity risk.

If we have P_t<<IR, then we can use the usual K-factor formula (4).

The notion of core-tail overbounding is very important theoretically because it justifies the use of overbounding methods when the tail is not fully known, and it justifies the use of Gaussian overbounding even when the tail is heavier than a Gaussian. However, in practice, such conditions on P_t (P_t<IR⁄n or P_t<<IR) are challenging to verify because IR is already very low (usually about 10^-7). This requires a fine knowledge of the error distribution to go far in the tail, which means a large experimental sample of values that is quite difficult to achieve. Thus, the core-tail overbounding technique is rarely used, and the assumption that the overbounding is valid up to infinity is often made. 

Excess Mass for Paired Overbounding 

In practice, constructing pairs of overbounds can artificially increase biases or variances on each of the overbounds to correctly bound the empirical distribution function. While this operation is often necessary, it inevitably leads to conservatism and larger protection levels than needed.

The left bounding distribution must have a heavier tail on the left (on the negative side) than the original error distribution, while at the same time have a lighter tail on the right. Because Gaussian distributions are often used for overbounding, this double condition is difficult to achieve and is often met at the price of very large biases. This specific problem with the pair overbounding technique can be partially resolved by the concept of excess mass overbounding [5] (excess mass CDF method), which considers a distribution mass greater than 1, typically 1+ ε where ε is referred as excess mass.

With such mass, the pseudo CDF is allowed to pass either beyond +1 or below -1. With this flexibility, we only need to ensure the properties of the overbound on one side of the empirical distribution, knowing the other side is no longer constrained. For Gaussian pair-overbounds, this adds a third degree of freedom to the variance and the bias.

Formally, we consider f, a pseudo-PDF that is positive and integrable, but we do not impose that its total probability weight is 1. Instead, we allow a total mass 1+ε larger than one (Equation 43).

The associated left pseudo-CDF is defined as F_L(x)=(1+ε)×F~_L(x), where F~_L is a regular CDF with total weight equal to 1. This means F_L(x) tends to 0 as the variable x approaches negative infinity as a regular CDF, but tends to 1+ε as x approaches infinity, relaxing the overbounding constraint by the left overbounding on the right.

For the right overbound F_R(x), the excess mass is applied on the survival function: 1–F_R(x)=(1+ε)×(1–F~_R(x)), where F~_R(x) is a regular CDF, which leads to F_R(x)=(1+ε)×F~_R(x)-ε. Consequently, the right pseudo-CDF goes to 1 as x approaches infinity like a regular CDF but tends to a negative value when x goes to negative infinity.

With this definition, the concept of pairs of overbounds can be expressed in the same terms and exhibits the same properties, particularly the stability under convolution (the proof can be reproduced identically as it uses only the positivity of the PDF), noting the mass of the sum of random variables is the product of the masses of each variable. Specifically, if the variables X and Y have respectively masses 1+ε_X and 1+ε_Y, then the sum X+Y has a mass of (1+ε_X)×(1+ε_Y). 

Definition: L_X and R_X associated to the pseudo-PDF f_L and f_R define a paired overbound with excess mass ε of the random variable X, and we note X⊆[L_X,R_X] if 

where F_L(x) and F~_R(x) are regular CDF (with total probability weight equal to 1).

Theorem 3: If X and Y are paired-overbounded with excess mass by a symmetric pair then the linear combinations are also paired-overbounded by a known formula. If X⊆[–R_X,R_X] and Y⊆[–R_Y,R_Y] then ∀(α,β)∈², αX+βY ⊆[–|α|R_X–|β|R_Y,|α|R_X+|β|R_Y]. (Here –R_X is defined as having the pseudo-PDF f_R(-x) and the sum and multiplication by a scalar are defined as for regular random variables). The total mass of the overbounding of the linear combination is (1+ε_X)×(1+ε_Y). Under these conditions, integrity in the pseudorange domain imply integrity in the position domain.

The proof is the same as the equivalent proof for paired-overbounding. With each addition, the total weight of the excess mass overbounding pair grows as it is the product of all masses involved.

Figure 5 illustrates paired overbounding with excess mass ε equals to 2.5×10^–2. The overbounds are two Gaussian with bias –/+ 0.3 and standard deviation 0.7 and the green area represents the domain that satisfies the overbounding of X by L_X on the left and R_X on the right. The QQ plot representation is not applicable to excess mass because the overbounds has total weight greater than 1. The allowed green zone is much larger with excess mass than in Figure 2, making the condition easier to verify.

Application to protection volumes: In the case of overbounding by Gaussian pseudo–PDFs, all the convenient addition properties of the Gaussian-distributed vector are maintained. All the formulas derived for the protection volume build with pair-overbounding remain valid, with only a modification in the K-factor. Explicit calculations show the new expression of the K-factor is

Because the complementary error function erfc is decreasing, we see K increases with ε. Thus, a larger excess mass makes it easier to pair-overbound the distribution, but leads to larger protection volumes.

When all the error sources i are paired-overbounded with excess mass by Gaussian distributions of standard deviation σ_i, bias ±μ_i and excess mass ε_i,the calculation of the protection volume remains particularly simple. It is given by

Where 

ξ is the multiplicative factor defined by Equation 16 and inflating K-factor defined by Equation 46.

Note the excess mass, as proposed in [5] requires the true tail is Gaussian or lighter to the Gaussian, because the overbound must hold up to infinite. It is interesting to combine the excess mass with core-tail approaches. We propose to include the excess mass only on the core overbound. In this case, we have for each error contributor:

By projecting Equation 47 in position domain, we get:

Equation 48 can be arranged as:

By inverting the equation, we get, in Equation 50, the new formulation for K-factor:

Two Steps Overbounding 

The objective of the approach proposed in [7] is to mix the two concepts of central CDF-overbound and paired overbound. From the empirical distribution of an unmodelled residual error, the objective is to construct a weaker form of paired overbound by Gaussian distributions (the Gaussian distribution is chosen for computational simplicity).

We start by constructing an intermediate pair-overbound where each bound is unimodal and symmetrical around its mean, but not necessarily Gaussian. We know (Theorem 3) this paired overbound is stable by convolution. We impose the properties of symmetry and unimodality because they are part of the hypotheses of the CDF-overbound. The second step is to find a Gaussian distribution (necessarily symmetrical and unimodal around their mean) where CDF-overbounds the left and right distributions. The assumptions of the CDF-overbound are met and imply these pairs of Gaussians overbounds are stable by convolution.

The result of this method are the two Gaussian variables

which have the following property:

The resulting properties of the two-step overbounding are weaker than those of the pair-overbounding, but they are sufficient to build a protection volume as follows:

Where K is the usual factor computed with Gaussian distributions (Equation 4).

This protection level is correct because of

The goal is to build a Gaussian two-steps overbound defined by μ_X, σ_X that is stable by linear combination, making it possible to perform the overbounding at range level and to build a protection volume at position level.

Theorem 5: If μ_X, σ_X define a two-steps overbound of X and μ_Y, σ_Y define a two-steps overbound of Y, then the linear combination Z=αX+βY can be two-steps overbounded by a pair of Gaussians with bias μ_Z=|α| μ_X+|β| μ_Y and variance

Under this condition, integrity in the pseudorange domain imply integrity in the position domain.

Proof:

The proof makes extensive use of previous results. Let us start with stability by convolution and consider Z=X+Y. Can we build a two-steps overbound of Z from the two-steps overbounds of X and Y?

We know the intermediate paired overbounding is stable by convolution, and this step requires no particular assumption on the underlying distributions of X and Y:

Then we use that the CDF-overbounding is stable by convolution. This is where the unimodality and symmetry around the median of the intermediate distributions is essential:

Because the CDF-overbounds are Gaussian, we have O_LX+O_LY~N(μ_LX+μ_LY, σ²_LX+σ²_LY) and similarly for O_RX+O_RY.

Finally, μ_Z=μ_X+μ_Y, we have μ_Z≥ max (|μ_LX+μ_LY|,|μ_RX+μ_RY|) and also σ²_Z≥ max(σ²_LX+σ²_LY, σ²_RX+σ²_RY). Because the tail probability for a Gaussian distribution is an increasing function of the standard deviation and the mean, the previous inequalities ensures the two Gaussian variables

of mean ±μ_Z and variance σ²_Z have the following property:

which is the essential property for the protection level to be correct. 

Note these properties on the tails of the distributions are necessary for the protection level formula to be correct but are not in themselves sufficient for a two-steps overbound because they are not stable by convolution. The stability by addition comes from the intermediate paired-overbound constructed in the first step.

The multiplication by a scalar poses no additional difficulties: if Z=αX then Z ⊆ [αL_X,αR_X] if α if positive (and Z ⊆ [αR_X,αL_X] for the negative case). In both cases αL_XαO_LX and αR_XαO_RX because the considered distributions are unimodal and symmetric around their means. We have αO_LX~N(αμ_LX,α² σ²_LX) and αO_RX~N(αμ_RX, α² σ²_RX) and the choice μ_Z=|α|μ_X and σ_Z=α² σ²_X ensures μ_Z≥max (|αμ_LX|,|αμ_RX|) and σ²_Z≥max (α² σ²_LX, α² σ²_RX). 

The two-steps overbounding construction allows for the transfer of integrity from the measurement domain to the position domain without specific assumptions of symmetry, centering and unimodality of the empirical error distribution by transmitting only two Gaussian parameters per line-of-sight (namely μ_X and σ²_X). This approach is less conservative than the paired overbound by two symmetric Gaussians because the needed property concerns only the left tail

The stability by linear combination is guaranteed by the existence of the intermediate symmetric and unimodal paired-overbound, but the explicit knowledge of the intermediate distribution is not needed for protection volume calculation.

The limiting aspect of the two steps overbounding method is the quality of the final overbound is very dependent on the construction of the first paired overbounds (first step). This step requires a complex optimization algorithm, with one such algorithm described in [7]. Another drawback of the method is it is difficult for a user with knowledge of the original error distribution and the parameters μ_X and σ²_X to verify the received parameters effectively form a correct two-steps overbounding of the error distribution without the information of the algorithm used in the first step.

Wide Sense CDF-Overbounding 

The approach of Weak CDF-over-bounding is introduced in [11]. Similar to the two-step overbound concept, the approach aims at taking advantage of both the CDF-overbounding and pair-overbounding definitions: the simplicity of the first one and the robustness of the second. The main idea is to impose a condition equivalent to CDF-overbounding for biased distribution, but without the assumptions of symmetry and unimodality. When performing linear combinations, the stability of the protection volume formula is lost because of the dropped assumptions. However, it is possible to quantify and bound the worst deviations from the Gaussian distribution and encapsulate them in a suitable inflation factor in the protection volume formula to ensure integrity.

In more precise terms, let X be a distribution (typically the distribution of an error contributor in a GNSS pseudorange measurement) with median b. We do not require X to be unimodal nor symmetric around its median. A weak CDF-overbound is given by a pair of Gaussians with mean ±μ_X and identical variance σ² such that μ_X≥|b| and furthermore:

The condition is identical to the second step of the two-steps overbounding method, but without the first step guaranteeing the stability of the property under linear combinations. If our quantity of interest is given by

where all X_i are individually weakly CDF-overbounded by Gaussian pairs of parameters μ_i and σ_i and n being the number of each error source, it is shown in [11] that the weak CDF-overbounding condition is sufficient to build a protection volume for E. The protection level formula is:

In this formula, K is the usual Gaussian K-factor (σ is the usual Gaussian standard deviation) where

and the new term A_n,K is an inflation factor that takes into account the weaker assumptions taken in the definition. It quantifies somehow the worst deviations from the Gaussian distribution if each term follows the weak CDF-overbound condition. The inflation factor A_n,K depends on the value of the K-factor (and the required integrity risk) but also on the number of terms in the linear combination. The factor A_n,K has no simple analytical formula, but a table of values can be pre-computed and used as such in a given context.

Note the parameter n is the number of independent error contributors and can be larger than the number of satellites in view if there are several error sources for each line-of-sight.

In this method, the properties of CDF-overbounding cannot be used directly because important assumptions on the unknown distribution are not met. However, the conditions of CDF-overbounding by a Gaussian are equivalent to a pair-overbound by two symmetric distributions, the left and right half-Gaussian. Furthermore, the properties of pair-overbounding do not use the assumptions of symmetry and unimodality. One can build a pair-overbound of the final error distribution as linear combinations of half-Gaussians. The last step is to build a pair-overbounding distribution for all coefficients in the linear combinations of half-Gaussians, such that the result is independent of the geometry of the problem. This allows conservative protection volumes to be built for a specific integrity risk. These protection levels are then divided by the usual K-factor and interpreted as an inflation factor A_n,K integrated in the protection volume formula. Note the factors presented in Table 1 and [11] are upper-bounds and future improvements on the method may reduce the numerical values of the inflation factors and thus of the protection volumes.

The main advantage of weak CDF-overbounding is its simplicity. The condition to verify for weak CDF-overbounding is of the same complexity as usual CDF-overbounding, but fewer assumptions on the underlying distribution are needed. The user only needs to store a pre-computed table of values of inflation factors (computed for the required integrity risk) to use when computing its protection volume. On the other hand, weak CDF-overbounding can lead to large protection volumes if the number of contributors is large and the K-factor is low. 

Conclusion 

This article shows the overbounding concepts that play a crucial role in demonstrating integrity are neither intuitive nor straightforward. However, the overbounding concepts using Gaussian overbounds are designed to keep the procedure as simple as possible. Indeed, the stability by linear combination of the overbounding properties and of the Gaussian distribution allows the user to manipulate the range errors as if they were Gaussian (by adding their standard deviations in quadrature) and apply the protection volume formula that mostly differ by the K-factor. From the integrity system point of view, the advantage is to focus on monitoring range error distribution and to send few parameters to the user, allowing for the construction of correct protection volumes. 

Table 2 summarizes the main assumptions and implications of the different concepts applied to Gaussian overbounds.

Despite extensive work on the subject, several open points remain on overbounding. 

First of all, all the studied concepts consider independence between all contributions to the positioning errors, and consequently between line of sights (Table 1). In practice, a correlation between the lines of sight could be caused by the troposphere residual errors, multipath or ODTS algorithm. [12] introduced a Power Spectral Density (PSD) overbounding concept that can guarantee integrity transfer of correlated Gaussian errors (with unknown and arbitrary correlation pattern), using a Gauss-Markov processes that overbound the PSD. To our knowledge, this concept is not used for single point positioning but is key for the integrity demonstration for Kalman filter. Generalization of this concept to non-Gaussian distributions will be a key step both for single point positioning and Kalman filtering.

Secondly, the concepts of overbounding presented are adapted to one dimensional quantities only. For example, the protection level formulas should be applied direction by direction for the positioning solution. A theory of overbounding multidimensional distributions is missing to, for example, build a protection volume for the norm of the horizontal positioning error. One difficulty of such a theory is the norm of a vector is not a linear combination of its components, and thus all the properties presented do not apply to this problem [13]. 

Last, the practical evaluation of overbounding remains a challenge because it involves estimating quantiles with low probability. This requires collecting and analyzing huge quantities of experimental data, which is costly, cumbersome and generally unrealistic. A promising alternative could be to extrapolate the tails of the distributions beyond the available data based on extreme value theory. 

Appendix

CDF OF THE SUM OF TWO INDEPENDENT RANDOM VARIABLES

We start by formulating the expression for the CDF of the sum X+Y. By definition of the CDF, F_(X+Y) (z)=P(X+Y≤z). The probability P(X+Y≤z) is obtained by integrating the join density function f_(X,Y) (x,y) of the couple of random variables (X,Y) on the set of x,y respecting the inequality x+y≤z or equivalently y≤z-x. As the random variables X and Y are independent, the joint PDF is the product of the individual PDFs:

The PDF of the sum of the two independent variables X and Y is obtained by making, in Equation 59, the change of variable y=v-x, interchanging the order of integration, and derivating the CDF with respect to z:

Thus, the density f_(X+Y) appear as the classic convolution product Equation 61 of the densities, noted as f_(X+Y)=f_X*f_Y:

The CDF of the sum of the two variables X and Y is obtained by recognizing in Equation 59 the CDF of the random variable Y:

Injecting F_Y (z-x) given by Equation 62 in the expression of F_(X+Y) (z) in Equation 59, we then get:

By making the integral of the joint density function respecting now the inequality x≤z-y in Equation 59, it comes:

Therefore, the CDF of the sum of the two variables X and Y is the convolution of the CDF of the one with the PDF on the other indifferently:

SUM OF TWO SYMMETRIC UNIMODAL RANDOM VARIABLES

In this section, we want to prove the following: if X and Y are two independent, symmetric and unimodal random variables, then their sum X+Y is also symmetric and unimodal [8]. 

Let f_X, f_Y, be the PDFs of X and Y and f_(X+Y)=f_X*f_Y the PDF of X+Y. Then f_(X+Y) is symmetric because, for all x∈R:

In this manipulation, we have used the symmetry of f_X and f_Y and the change of variable t→-t.

Let us now fix two reals 0≤a≤b and prove that f_(X+Y) (a)≥f_(X+Y) (b). We have from the ordering of a and b that for all x, |x-a|≥|x-b| if x≥((a+b))⁄2 and |x-a|≤|x-b| if x≤((a+b))⁄2. From the unimodality and symmetry of Y, this implies that f_Y (x-a)≤f_Y (x-b) if x≥((a+b))⁄2 and the contrary if x≤((a+b))⁄2. Similarly, we have, for x≥((a+b))⁄2, |x|≥|x-a-b| and so f_X (x)≤f_X (x-a-b) and also, for x≤((a+b))⁄2, f_X (x)≥f_X (x-a-b).

This shows that for all x∈R, we have:

Because both terms are positive for x≤((a+b))⁄2, else both terms are negative.

Integrating the positive product defining by Equation 66 on x, and using the symmetry of f_X,f_Y we get:

And using the symmetry of f_(X+Y), right side of inequation 67 becomes:

Thus f_(X+Y) is decreasing for positive x, which proves the unimodality of X+Y.

A COUNTER EXAMPLE TO A NAÏVE APPROACH 

For a given integrity risk IR and associated Gaussian K-factor, it is tempting to check integrity at the pseudorange level by making sure that for all lines of sight:

However, the following example demonstrates this approach is not sufficient to guarantee the integrity at position level. The reason is this condition tests a certain quantile of the pseudorange error distribution, whereas the stability by linear combinations require criteria to be met across the entire distribution.

Let us use a numerical counterexample. We first take a geometry matrix (here corresponding to 7 GPS satellites above Toulouse  on September 3, 2002, 0h00 in ENU frame)

Let us consider the case where all error variances on the pseudorange errors σ_i^2 are equal to 1. Then we build the S matrix as S=(G^T G)^(-1) G^T. The third line of S is

The vertical protection volume VPL has radius given by Equation 69

with K_V=5.33 and σ_V=√(S_(3,i)^2 σ_i^2 )

If the vector of residual errors is

then each line of sight passes the integrity test (meaning that ∀i,b_i<K_V σ_i=5.33), but the vertical error is

which exceeds the protection level radius.

This counter-example does not question the principle of integrity transfers from pseudo distance to position, but the fact that it can be done by the sole definition of confidence intervals in the pseudo distance domain. To establish the equivalence between both realms, the property of overbounding must be verified across the entire distribution, at least up to the desired quantile.

Indeed, if the overbound is applicable for each line of sight, it is extremely improbable to obtain the vector  proposed in the example. In addition, if we only know the inequality is satisfied for each line of sight, we have no information about the probability of occurrence of the given vector  in the example, even though it leads to a lack of integrity in the domain of positions.

References 

[1] Minimum operational performance standards for Global positioning system/ wide area augmentation system airborne equipment, DO-229, RTCA ed. Washington, DC.

[2] B. DeCleene, Defining pseudorange integrity – Overbounding, In Proc. 13th Int. Techn. Meeting Satellite Div. Inst. Navigat., Salt Lake City, UT, USA, Sep. 2000, pp. 1916–1924.

[3] J. Rife , S. Pullen, B. Pervan, and P. Enge, Paired Overbounding and Application to GPS Augmentation, Proceedings IEEE Position, Location and Navigation Symposium, pp. 439-446, July 2004.

[4] J. Rife, S. Pullen, B. Pervan, and P. Enge, Paired Overbounding for Nonideal LAAS and WAAS Error Distributions, IEEE Trans. Aerosp. Electron. Syst., vol. 42, no. 4, pp. 1386–1395, Oct. 2006.

[5] J. Rife, J. Blanch and T. Walter, Overbounding SBAS and GBAS Error Distributions with Excess-Mass Functions, in Proceedings of the GNSS 2004 Internat. Symp. On GNSS/GPS, Sydney, Australia,6-8, Dec. 2004.

[6] J. Rife, S. Pullen, B. Pervan, Core Overbounding and its Implications for LAAS Integrity, Proceedings of the 17th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2004), Long Beach, CA, Sept. 2004, pp. 2810-2821.

[7] J. Blanch, T. Walter and P. Enge, Gaussian Bounds of Sample Distributions for Integrity Analysis, IEEE Trans. Aerosp. Electron. Syst., vol. 55, no. 4, pp 1806-1815, Aug. 2019.

[8] M. Earnest, link.

[9] J. Antic, O. Maliet, and S. Trilles, “SBAS Protection Levels with Gauss-Markov K-Factors for Any Integrity Target”; NAVIGATION: Journal of the Institute of Navigation September 2023, 70 (3) navi.594

[10] K. Mimouni, O. Maliet, J. Antic, “A simple and robust K-factor computation for GNSS integrity needs”. ION plan, pp 399-407, 2023

[11] Maliet, O., Mimouni, K., Antic, J., & Trilles, S. (2025). Wide-Sense CDF overbounding for GNSS integrity. NAVIGATION: Journal of the Institute of Navigation June 2025, 72 (2) navi.697; link

[12] Langel, S., Crespillo, O. G., & Joerger, M. (2020, April). “A new approach for modeling correlated Gaussian errors using frequency domain overbounding”. In 2020 IEEE/ION Position, Location and Navigation Symposium (PLANS) (pp. 868-876). IEEE.

[13] I. Nikiforov, “From pseudorange overbounding to integrity risk overbounding”, NAVIGATION, Vol 66, Issue 2, Summer 2019, pp 417-439.

[14] Z. W. Birnbaum, “On Random Variables with Comparable Peakedness”, The Annals of Mathematical Statistics, 19 (1), pp 76-81 doi:10.1214/aoms/1177730293

Authors

Julie Antic is a specialist in GNSS integrity algorithms and performances at Thales Alenia Space in Toulouse, France. She holds a Ph.D. in probability and Statistics from Paul Sabatier University, France, as well as an engineering degree in applied Mathematics from INSA in Toulouse, France. Her areas of activity include advanced GNSS augmentation systems for high accuracy and integrity, advanced receiver autonomous integrity monitoring and overbounding concepts.

Odile Maliet graduated from École Polytechnique and received her Ph.D. degree in Macroevolution from École Normale Supérieure (ENS), Paris in 2018. Between 2018 and 2020, she worked as a postdoc on the use of Bayesian techniques on phylogenetics empirical data at ENS, Paris. Since 2021 she has worked on integrity concepts in Advanced Projects at the Performance and Processing Department of Navigation Domain, Thales Alenia Space.

Kin Mimouni graduated from École Polytechnique, Paris, and received his Ph.D. in Theoretical Physics from École Polytechnique Fédérale de Lausanne (EPFL) in 2019. Since 2021 he has worked on GNSS algorithms and integrity concepts, first as a post-doc in the Télécommunications Spatiales et Aéronautiques (TéSA) laboratory in Toulouse, and since 2023 as an engineer in Advanced Projects at the Performance and Processing Department of Navigation Domain France, Thales Alenia Space.

Sébastien Trilles is an expert in orbitography and integrity algorithms at Thales Alenia Space in Toulouse, France. He holds a Ph.D. in Pure Mathematics from the Paul Sabatier University and an advanced master’s degree in Space Technology from ISAE-Supaero. He heads the Performance and Processing Department where high precise algorithms are designed as orbit determination, clock synchronization, time transfer, reference time generation, integrity and ionosphere modeling algorithms for GNSS systems and augmentation.

The post Working Papers: Overview of Overbounding Techniques for Integrity Purposes appeared first on Inside GNSS - Global Navigation Satellite Systems Engineering, Policy, and Design.

New research supported by the European Space Agency’s (ESA) Third Party Missions programme has generated Arctic-wide sea ice freeboard maps using GNSS-Reflectometry (GNSS-R) data captured by Spire Global, Inc.’s GNSS-Reflectometry (GNSS-R) multipurpose listening constellation.

Led by the Technical University of Munich (DGFI-TUM) and the Norwegian Research Centre, the study leveraged Spire’s grazing-angle GNSS-Reflectometry (GNSS-R) — a radio frequency (RF) sensing technique that analyzes reflected navigation signals — to retrieve sea ice freeboard measurements across an entire winter season. The results show strong alignment with established altimetry datasets, including ESA’s CryoSat mission, validating the complementary role of commercial satellite data alongside government missions.

While GNSS signals have long been used for positioning, this research highlights how reflected signal analysis can extend their value into large-scale Earth observation applications, delivering persistent coverage independent of sunlight or weather conditions.

“Advances in miniaturization, digital signal processing, and machine learning have fundamentally changed what’s possible in RF sensing,” said Theresa Condor, Chief Executive Officer of Spire Global. “Commercial constellations can now deliver persistent, high-quality RF data that complements traditional government systems with greater flexibility and cost efficiency. As environmental monitoring requirements intensify, we’re seeing agencies increasingly integrate commercially sourced RF datasets into operational architectures, reflecting the continued maturation of this market and the growing role of commercial infrastructure in government missions.”

Read more on the research from ESA: Reflected satellite signals unlock new insights into Arctic sea ice

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Over the last several days, maritime analytics providers have documented interference events affecting more than 1,000 ships in the Middle East Gulf, alongside a growing pattern of AIS anomalies and “dark” operations. At the same time, tanker and container traffic has slowed or stopped near the Strait of Hormuz, and leading war-risk insurers are withdrawing cover for the region. 

The episode illustrates in practical terms what a contested RF environment means for ships that still rely heavily on satellite-derived position for navigation, tracking and compliance.

Interference profile: GPS jamming and AIS spoofing on a regional scale

Maritime intelligence firm Windward reports that more than 1,100 vessels experienced GPS and AIS interference across the Middle East Gulf within a single 24-hour period following the outbreak of hostilities between Iran, the United States and Israel. Ships’ reported positions were displaced onto airports, inland locations in Iran and the Gulf states, and even over a nuclear power plant, producing track histories that are clearly inconsistent with physical reality. 

A parallel assessment reported by Wired, based on analysis of satellite navigation attacks since the start of the air campaign against Iran, arrives at a similar figure of roughly 1,100 ships affected, underscoring that interference is not limited to a small subset of vessels or a single narrow area. 

Dryad Global notes “heightened risk of GPS jamming and AIS spoofing” in the Gulf of Oman and Strait of Hormuz, explicitly linking recent anomalies to Iranian naval exercises and electronic warfare activity. 

Taken together, the data suggests:

• GNSS-derived position can become systematically biased over wide areas, not only momentarily lost.
• AIS tracks based on those positions may show vessels apparently transiting over land, clustered around inland targets, or moving in circular or jagged patterns that reflect repeated loss and reacquisition of signal.
• Some operators respond by switching AIS off altogether, which protects them from misinterpretation of spoofed positions but reduces visibility for collision-avoidance and traffic management.

From a PNT standpoint, this is a textbook case of how GNSS jamming and spoofing propagate through downstream systems that treat satellite position as authoritative.

Shipping, insurance and security advisories

The interference is occurring against the backdrop of a broader shipping disruption centered on the Strait of Hormuz.

Reuters reports that around 150 ships, including oil and LNG tankers, are currently stranded near the Strait of Hormuz, with at least five tankers damaged and crew casualties following drone and missile attacks. In the wake of US and Israeli strikes, Iran has announced that it is closing the strait, and many market participants now characterize conditions as a “de facto” closure of a route that normally carries about one-fifth of global oil exports and substantial volumes of gas.

In response to the increased risk:

• Major war-risk underwriters, including Gard, Skuld, NorthStandard, the American Club and others, are cancelling war-risk cover for ships operating in Gulf and Iranian waters from early March, with premiums for any residual cover rising sharply. 
• Container carriers such as Maersk and CMA CGM have begun rerouting or suspending services that would normally pass through Hormuz, adding to the reduction in commercial traffic through the area. 

On the governmental side, a recent advisory from the US Maritime Administration designates the Strait of Hormuz, Persian Gulf, Gulf of Oman and parts of the Arabian Sea as an area of active military operations and potential retaliatory strikes by Iranian forces. The advisory highlights the risk of hailing, boarding or detention of commercial vessels and directs operators to closely monitor updates and guidance from US Naval Forces Central Command. 

Although these notices are primarily focused on kinetic threats, several security circulars from P&I clubs and risk advisers now explicitly call out the likelihood of GPS interference and AIS anomalies in the region and recommend that ships treat GNSS-based position with caution when operating there. 

Implications for PNT resilience

The current pattern of events around Hormuz reinforces several points that have been discussed in standards bodies and industry forums for some time:

• GNSS reliability is not uniform. In certain strategic waterways, including parts of the Gulf and Strait of Hormuz, interference can reach a level where satellite-based positioning should be treated as advisory rather than authoritative. 
• Spoofed or displaced positions can have regulatory and commercial consequences, not just navigational ones, when automated compliance systems interpret false AIS tracks as evidence of port calls or territorial incursions. 
• “Going dark” on AIS reduces exposure to mis-located tracks but increases dependence on radar and visual watchkeeping, especially in confined waters.

For PNT system designers and policy-makers, the current situation underscores the value of alternative and complementary positioning sources, whether that means terrestrial systems, inertial aids, or hardened multi-constellation receivers, and the need to assume that in some regions, GNSS degradation will not be an exception but a recurring operating condition.

In that sense, the developments around Hormuz are less an isolated crisis than another data point in an evolving pattern: satellite navigation has become a routine instrument in regional competition, and maritime navigation practices are having to adjust accordingly.

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The honors include:

• Best Industry Coverage for the PNT Leadership Summit issue and proceedings
• Best Subject-Related Package for Assured PNT: A National Imperative

Together, these recognitions acknowledge sustained, technically grounded work addressing PNT resilience as both an engineering challenge and a national infrastructure priority—examining threats, architectures, governance, and operational accountability across civil, commercial, and defense domains.

Established in 1955, the Jesse H. Neal Awards are widely regarded as the Pulitzer Prizes of the B2B media industry, honoring journalism that serves professional communities with rigor and clarity.

This acknowledgment belongs to the Inside GNSS community. We thank our contributors, advisors, and readers whose expertise, candor, and engagement made this work possible.

The post A Shared Recognition for the Inside GNSS Community appeared first on Inside GNSS - Global Navigation Satellite Systems Engineering, Policy, and Design.

The collaboration aims to fuse satellite imagery, real-time sensor feeds and onboard computer vision into navigation and targeting solutions that remain reliable when GPS and other GNSS signals are unavailable. 

Under a Letter of Intent, the companies plan to integrate CGI’s AI, edge computing and visual analytics platforms with Vantor’s Tensorglobe spatial intelligence environment and Raptor visual navigation software. Tensorglobe is Vantor’s AI-powered spatial intelligence platform that unifies data from satellites, drones and ground sensors into what the company describes as a “living 3D replica of Earth.” Raptor is a vision-based positioning suite that uses onboard cameras and 3D terrain data to estimate aerial position and extract ground coordinates in real time, reducing dependence on GNSS for unmanned systems and other platforms. 

Fusing Visual Navigation and Spatial Intelligence

According to the joint announcement, CGI will bring its Machine Vision and SignalSense technologies, along with secure access to satellite data and space-based sensing, while Vantor contributes Tensorglobe’s data fusion and Raptor’s GNSS-resilient navigation capabilities. The goal is to deliver integrated products that provide precise location data, real-time analytics and operational navigation in GNSS-denied or signal-degraded areas, supporting faster decision-making in complex environments. 

On the navigation side, integrating Raptor into the joint offering is intended to help users maintain accurate positioning and coordinate generation when GNSS is jammed, spoofed or unavailable—using visual matching against high-resolution 3D terrain models instead of satellite ranging alone. On the intelligence side, Tensorglobe’s ability to orchestrate multi-constellation satellite imagery, airborne ISR and ground sensors into a common operating picture is expected to feed CGI’s AI analytics, supporting mission planning, dynamic monitoring and threat detection. 

CGI and Vantor say they will initially target defense, national security and civil government customers in the UK, Europe and allied markets that need AI-enabled edge computing and space-based situational awareness for operations with poor or zero GNSS coverage. The companies also frame the alliance as a way to deliver interoperable and sovereign solutions, aligning with European and allied priorities around resilient PNT, sovereign data handling and reduced dependence on single-source infrastructure. 

For GNSS users and system integrators, the partnership fits into a broader trend: using spatial intelligence, visual positioning and alternative signals to harden PNT architectures against jamming, spoofing and outages, while feeding richer context into command-and-control and autonomy stacks rather than treating navigation and ISR as separate problems.

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JUDAH LEVINE, UNIVERSITY OF COLORADO, PASCALE DEFRAIGNE, ROYAL OBSERVATORY OF BELGIUM, ILARIA SESIA, ITALIAN METROLOGY INSTITUTE, GIULIO TAGLIAFERRO, INTERNATIONAL BUREAU OF WEIGHTS AND MEASURES, MICHAEL WOUTERS, NATIONAL MEASUREMENT INSTITUTE

Coordinated Universal Time (UTC) has been recommended as the unique time scale for international reference time stamps and is the basis for civil time in most countries [1]. Time zones, which are established by local administrations, are defined by an offset from UTC. Some applications are required to use time stamps based on UTC either by regulation or by statute [2-4]. There are advantages to the use of time stamps based on UTC, even when it is not required to do so, because this facilitates combining data from multiple sources or when international coordination is important.

Time signals from global navigation satellite systems (GNSS) are widely used as the reference time in many applications, and it is important to understand the requirements that ensure GNSS time stamps are traceable to UTC from both a technical and a regulatory perspective [5]. This article describes how UTC is defined and realized and how a prediction of UTC is included in GNSS data transmissions.

The Definition and Realization of UTC 

The UTC time scale is a paper time scale that has no physical realization. It is computed monthly by the International Bureau of Weights and Measures (BIPM) based on data from several hundred atomic clocks located at National Metrology Institutes (NMIs) and other time centers in various countries. Many laboratories operate local ensembles of atomic clocks and use the data from these ensembles to compute and disseminate a local UTC estimate. This local estimate is identified as UTC(k), where k is the acronym for the laboratory. The estimate of UTC computed by the U.S. Naval Observatory (USNO) is UTC(USNO) and the estimate computed by the National Institute of Standards and Technology (NIST) is UTC(NIST). 

The computation of UTC for any month is published in BIPM Circular T by the tenth day of the following month [6]. This circular tabulates UTC-UTC(k) every five days for every participating laboratory. A rapid version of UTC, called UTCr [7], is also published by the BIPM every Wednesday. It lists daily values of UTCr-UTC(lab) through the previous Sunday. These data are published on the BIPM website and are distributed by email (Figures 1 and 2). 

GNSS Time Signals

The system time of a GNSS constellation, GNSS_T, is generated by the ground segment from an ensemble of clocks located on the ground at the control center and tracking stations. It also can include the clocks in the satellites [8-11]. Each satellite in a constellation transmits a prediction of the offset between the time of the clock on the satellite and the system time of the constellation, which is uploaded to the satellites periodically. GNSS constellations also broadcast bUTC_GNSS, a prediction of the difference between GNSS_T and UTC (including a 3-hour offset for the GLONASS system) that is derived from the UTC prediction of timing laboratories. This prediction is transmitted in two parameters: an integer giving number of whole seconds difference between UTC and the GNSS system time, and a fractional part, which specifies the difference modulo 1 s. The first parameter changes only when a leap second is inserted into UTC and not at other times. (The GLONASS constellation uses UTC as the system time so only the fraction is transmitted in the navigation message.)

For the GPS constellation, this prediction is derived from UTC(USNO) maintained at the U.S. Naval Observatory. The GLONASS constellation broadcasts a prediction based on UTC(SU), which is realized at the Russian Metrology Institute of Technical Physics and Radio Engineering (FSUE, VNIFTRI). The Galileo constellation uses a prediction derived from a collaboration of five European National Metrology Institutes. The BeiDou system uses UTC(NTSC) realized at the National Time Service center of China and UTC(NIM) realized at the China National Institute of Metrology. Regional systems also broadcast similar messages. The formats of the respective messages are GNSS-specific and are documented in the respective Interface Control Documents.

The Role of the BIPM

In addition to computing UTC and publishing the differences of UTC-UTC(k), the BIPM evaluates the difference between UTC and the predictions of UTC broadcast by the various GNSS constellations, bUTC_GNSS. These differences are published in Section 4 of BIPM circular T [6]. This section lists the difference in ns between UTC computed by the BIPM and the prediction of UTC transmitted by the GPS, GLONASS, Galileo and BeiDou satellites for every day in the monthly reporting period for that issue of Circular T. Table 1 shows the values from the August 2025 editor of Circular T interpolated from the five-day reporting interval in Circular T to a daily value at 0 UTC [12]. 

Linking User Equipment to UTC

There are two common configurations that support a link between the user’s time reference and UTC(k) or UTC. In the first configuration, the user has a clock (or an ensemble of clocks) that provides the reference signal for a GNSS timing receiver. The receiver does not discipline the free-running local clock (or ensemble) in the short term, but measures its time with respect to the signal broadcast by the satellites of some constellation. These data are combined with the data in the navigation message to (1) correct for the transit time between the satellite and the receiver, (2) include the offset between the satellite clock and the GNSS system time, and (3) add the prediction of the offset between the system time and UTC. 

Most receivers can be configured to implement these calculations in firmware, and the output data gives the difference between the local reference and the broadcast prediction of UTC. The signal from this clock (or clock ensemble) can be used in the user’s application or the application’s clock can be compared to it. The system connected to the GNSS receiver may be completely free-running and not disciplined by the GNSS data; its offset is recorded and used to adjust the downstream data. In some configurations, the time or frequency of the local reference clock is adjusted from time to time so the measured time difference is kept within some administratively defined tolerance. The interval between adjustments depends on this tolerance and on the frequency stability of the local reference, and it can range from minutes for a rubidium-based reference to hours or days for a cesium-based device. 

The second configuration, which is much more common, combines a GNSS receiver and an oscillator in a single device. There are many commercial systems that realize this configuration and often provide several outputs (5 MHz, 10 MHz and 1 pps) that are disciplined by the data received from the GNSS constellation. The simpler systems use the code data transmitted on the L1 frequency, but dual-frequency receivers and more sophisticated carrier-phase analyses are possible. The first GNSS disciplined oscillators usually used signals from the GPS constellation, but newer systems can track satellites from more than one constellation simultaneously. The output signal might be based on only the satellites from one constellation at any time or on a combination of the data received from multiple constellations. Either solution can produce significant steps in the output signals, especially in the PPS data, when the reference constellation changes. The details of the disciplining algorithm are often proprietary; the output could be disciplined to GNSS system time or to the prediction of UTC, and traceability to UTC would require the additional adjustment that incorporated the data published in BIPM Circular T. This additional adjustment, based on data from Circular T, may be small enough to ignore in some applications. 

The first configuration is more flexible; the measurement process can accept data from multiple sources, including common-view data, and implement more sophisticated post-processing methods. The adjustment process for the local reference clock can be adjusted to meet the requirements of the user’s application. The second configuration, on the other hand, may provide adequate performance in many applications. It is much simpler to operate, and this simplicity may be the deciding factor for many users.

Combining Data from Multiple Constellations

Combining GNSS signals from multiple constellations can significantly improve the timing performance of a user’s receiver, especially in locations with limited visibility of the sky. This approach requires a knowledge of the offsets between different GNSS time scales, which are at the level of a few ns and vary in time. It is possible for a user to solve for the inter-system bias between constellations [13] if enough satellites from both constellations are visible at the same time, but this is not always the case, and the broadcast values must be used [14]. The broadcast of the predicted time difference between each GNSS system time and UTC greatly simplifies the job of combining signals from multiple constellations when only broadcast data are available. The use of UTC as the common reference time scale eliminates the need for maintaining multiple inter-system bias values.

Metrological Traceability

The International Vocabulary of Metrology (VIM) defines traceability as the “property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty” [15]. The International Telecommunications Union (ITU) [16] and the International Laboratory Accreditation Conference adopted the same definition and refer to the International Organization for Standardization (ISO/IEC) standard 17025 [17].

The signals transmitted by the GNSS constellations can be traceable to UTC. The broadcast signals are linked to UTC through the UTC(k) data of a timing laboratory, and the transmissions are monitored by the BIPM with results published in Circular T. A user can also establish a common-view relationship with a timing laboratory, which can provide a near-real-time estimate of the offset of the user’s timing system with respect to the UTC(k) of that laboratory. In either case, these data are processed by the receiving system at the user’s site, and the calibration and statistical characteristics of this system directly affect the accuracy and stability of the timing data that control the user’s application. 

Timing Receiver Calibration

The calibration of the user’s equipment that is required to complete the demonstration of traceability should be performed by a method validated by a National Metrology Institute or a Designated Institute that participates in the Mutual Recognition Agreements (MRA) and have their Calibration and Measurement Capabilities (CMC) published in the Key Comparison Database maintained by the BIPM [18]. The Key Comparison Database maintains the equivalency between different organizations and guarantees the international acceptance of calibrations performed by different agencies that participate in the MRA.

The manufacturer’s published specifications can be the basis for specifying the performance of a stand-alone GNSS receiver or a GNSS receiver and disciplined oscillator combination. These specifications should be based on a calibration of one example of a particular model with some allowance for the variation among devices of the same type. For applications that require only modest accuracy (not greater than about 1 µs) a statement by the manufacturer that the device has been type-approved to that level is sufficient. The transmission delay even through a very long antenna cable is unlikely to invalidate this assumption.

The use of “type approval” may not be adequate for applications that require sub-microsecond accuracy. One method of calibrating a receiving system is to compare the output of the system to a source of UTC(k) by using a time-interval counter to monitor the difference. This method depends on an independent source of UTC(k), which might be provided by a traveling calibrated receiver, by transporting a running clock from the UTC(k) source to the location of the receiver, or by operating the device to be calibrated at a laboratory where a source of UTC(k) is available. 

Using a traveling receiver is the best choice because it tests the receiving hardware with the antenna and cables in the environment where it will be used. The BIPM uses this option to calibrate the time-transfer equipment at timing laboratories. Although the traveling receiver is calibrated, it is operated at a location where its position may not be known accurately, and where it may be influenced by multipath effects that are not the same as the effects on the device under test. These considerations are usually not important unless the required accuracy must be better than about 50 ns.

A calibration based on carrying a running clock from the source of UTC(k) to the location of the user’s equipment is feasible if the distance is not too great and the required calibration accuracy is not too high. For example, the GPS receivers at the NIST radio station in Fort Collins, Colorado, are calibrated by carrying a rubidium oscillator from the source of UTC(NIST) in Boulder to Fort Collins, a distance of about 100 km by road. The calibration is repeated by carrying a calibrated GPS receiver between the two locations. The accuracy of either method is estimated to be about 15 ns, and the two methods agree within this uncertainty.

Transporting the device under test to a location where an independent source of UTC(k) is available is usually the most difficult solution. It may be impractical to disconnect the antenna cable at the site, so a different cable must be used for the calibration. The delay through the actual cable can be estimated with a time domain reflectometer, but this method tests the cable with signals that are not the same as signals from real satellites. The impedances at the end-points are also different. 

If the system to be calibrated provides the contribution of each satellite in view to the composite output, then the common-view method can be used to calibrate the receiver. (Unfortunately, many disciplined oscillators do not provide these data.) The system to be calibrated measures the difference between the local clock (or clock ensemble) and the system time of the constellation by using the data from each satellite in view. These data are compared, satellite by satellite, with the same measurements made at a location where UTC(k) is available. The common-view difference cancels or attenuates the contributions of the satellite clock and the orbital parameters, which are common to both data sets and cancel in the differences in first order. If the distance between the locations of the user and the UTC(k) laboratory is not too great, the contribution of the ionosphere may also be common to both measurements and cancel in the difference. A multiple-frequency measurement, which can correct for the contribution of the ionosphere, may not offer a significant improvement over a simple L1 comparison in this configuration, because the contribution of the ionosphere will be cancelled or attenuated in the common-view subtraction. The common-view method can operate continuously, and can also monitor the stability of the remote system.

Frequency Calibration

The techniques described for timing calibration can also calibrate the output frequency of the user’s system. A frequency calibration can be easier to realize than a time calibration because the absolute values of the delays in the equipment are not important, only the stability of these delays is. The stability of the output frequency of a quartz oscillator may be degraded by fluctuations in the ambient temperature, and the frequency estimated with a reference based on a rubidium or cesium device may be degraded by changes in the multipath contribution.

Specific Recommendations

The documentation from the manufacturer is the best source of information about a particular device. The following specifications provide general guidance on the methods to establish traceability [19].

1. If the application can accept a fractional frequency uncertainty of 10^-8 or greater with an averaging time of one day, or a time uncertainty of 1 µs or greater, then a certificate by the manufacturer that at least one unit of the model satisfies the requirement is adequate to establish traceability at this level. (A receiver used only as the reference for a server that supports NTP, the Network Time Protocol, may not require calibration, because the accuracy and stability of the NTP service is usually limited to not better than about 1 ms by the characteristics of the network connection between the server and the client systems.) 

2. If the application requires a fractional frequency uncertainty between 10^-8 and 10^-10 with an averaging time of one day or a time uncertainty between 100 ns and 1 µs, then the manufacturer should provide a certificate with every unit that satisfies the requirement. The manufacturer could validate the performance of each unit by comparing its output with a calibrated reference unit maintained at the manufacturer’s facility. 

3. If the application requires a fractional frequency uncertain of less than 10^-10 with an averaging time of one day or a time uncertainty of less than 100 ns, then the calibration can be challenging and should be performed at the user’s facility, if possible. 

4. If the application requires a fractional frequency stability of less than 10^-12 or a time uncertainty of less than 50 ns, then the calibration should be repeated periodically or the performance of the system should be monitored by common-view or an equivalent technique, which will require a dedicated GNSS timing receiver at the user’s site. The contributions of multipath reflections and the sensitivity of the equipment to fluctuations in the ambient temperature may be important. The impact of multipath reflections can be minimized by locating the antenna so it has an unobstructed view of the sky, and by using a directional “choke ring” antenna, which attenuates signals coming from low elevations. The sensitivity to fluctuations in the ambient temperature may be a problem if the local reference device is a simple quartz oscillator or if a long antenna cable is exposed to direct sunlight. 

It is important to maintain documentation that validates the traceability of any system. Configurations that support this capability are particularly useful.

Summary and Conclusion

Applications that use the timing data from GNSS systems often require legal and technical traceability to UTC. Even when traceability is not legally required, maintaining traceability to UTC simplifies combining the data from multiple constellations. The signals transmitted by GNSS systems are monitored by the BIPM and can be made traceable to UTC by the methods discussed. Ensuring the traceability of the timing data in a user application also depends on a calibration of the receiving equipment. The methods for realizing this calibration were presented and specific recommendations provided. Maintaining adequate documentation is important, and configurations that support real-time monitoring and log files are particularly useful. 

References 

(1) Conference generale des poids et mesures (CGPM) 2018 Resolution 2 of the 26th CGPM (2018), on the definition of time scales (https://bipm.org/en/committees/cg/cgpm/26-2018)

(2) MiFiR RTS 25: https://ec.europa.eu/finance/securities/docs/isd/mifid/rts/160607-rts-25_en.pdf

(3) Finra Rule 6820: https://www.finra.org/rules-guidance/rulebooks/finra-rules/6820

(4) IEEE Standard for Synchrophasor Measurement for Power Systems, IEEE C37.118.1-2011. https://standards.ieee.org/ieee/C37.118.1/4902.

(5) Dimetrios Matsakis, Judah Levine, and Michael Lombardi, Metrological and Legal Traceability of Time Signals, Inside GNSS, March/April 2019, pp. 48-58.

(6) https://www.bipm.org/en/time-ftp/circular-t

(7) https://www.bipm.org/en/time-ftp/utcr

(8) GPS system time: https://www.gps.gov/applications/timing

(9) Galileo system time: https://www.gsc-europa.eu/GST

(10) GLONASS system time: https://www.unoosa.org/documents/pdf/icg/2020/GLONASS_Time_2017_E.pdf

(11) BeiDou system time: http://en.beidou.gov.cn/SYSTEMS/Officialdocument/202001/P020231201549662978039.pdf

(12) https://webtai.bipm.org/ftp/pub/tai/other-products/notes/explanatory_supplement_v0.8.pdf

(13) G. Huang, Q. Zhang, W. Fu and G. Guo, GPS/GLONASS time offset monitoring based on combined precise point positioning approach, Advances in Space Research, Vol. 55, number 12, 15 June 2015, pp. 2950-2960. DOI: https://doi.org/10.1016/j.asr.2015.03.003. See also references in that text.

(14) GPS-Galileo Time Offset (GGTO): https://www.unoosa.org/documents/pdf/icg/2017/wgd/wgd4-2-2.pdf

(15) https://www.bipm.org/documents/20126/54295284/VIM4_CD_210111c.pdf

(16) ITU-R, TF-686-3, Glossary and Definitions of Time and Frequency Terms p 16. https://www.itu.int/dms_pubrec/itu-r/rec/tf/r-rec-tf.686-3-201312-i!!pdf-e.pdf

(17) SO 17025:2017, General requirements for the competence of testing and calibration laboratories, https://www.iso.org/ISO-IEC-17025-testing-and-calibration-laboratories.html

(18) https://www.bipm.org/en/cipm-mra

(19) P. Defraigne, J. Achkar, M. J. Coleman, M. Gertsvolv, R. Ichikawa, J. Levine, P. Uhrich, P. Whibberley, M. Wouters and A. Bauch, Achieving traceability to UTC through GNSS measurements, Metrologia, vol. 59, Number 6, October 2022. Metrologia, 59, 064001. DOI: 10.1088/1681-7575/ac98cb.

Authors

Judah Levine is on the faculty of the Department of Physics at the University of Colorado at Boulder. He recently retired from the Time and Frequency Division of NIST, where he worked on time scales and methods of distributing time and frequency information. He is continuing those projects at the University and is also a member of committees of the International Bureau 
of Weights studying the future of Coordinated Universal Time and possible time scales for the Moon.

Pascale Defraigne obtained her Ph.D. in Geophysics in 1995 at the Université Catholique de Louvain. Since 1997, she has managed the time and frequency activities at the Royal Observatory of Belgium, where the Belgian reference UTC (ORB) is maintained. Her research activities mainly concern the use of satellite navigation systems for time and frequency transfer. Pascale presently chairs the CCTF working group on GNSS time transfer, and contributes to the validation of Galileo timing signals.

Ilaria Sesia is a Senior Researcher and Head of the Time and Frequency Department at INRiM, where she works on time transfer, atomic clocks and time scales for satellite applications. Since 2004, she has been deeply involved in the design and development of the timing aspects of the Galileo System.

Giulio Tagliaferro received his Ph.D. in 2021 from Politecnico di Milano on precise GNSS measurement adjustment. He is currently a physicist at BIPM, where he works on GNSS time-transfer activities and receiver calibration supporting the realization of UTC.

Michael Wouters leads the time and frequency group at the National Measurement Institute in Sydney, Australia. His research focuses on using low-cost GNSS receivers for time-transfer. He chairs the Consultative Committee on Time and Frequency’s task group working on the traceability of GNSS timing signals to UTC.

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