Low-Earth orbit (LEO) systems have emerged as a promising complement to GNSS, offering higher received power, better satellite geometry and broader spectrum options. Researchers aim to evaluate whether LEO-PNT can complement or enhance GNSS performance through large-scale simulations and design comparisons.
Researchers from Tampere University and Universitat Autònoma de Barcelona published (DOI: 10.1186/s43020-025-00186-5) a comparative analysis in the December 2025 issue of Satellite Navigation. The study investigates how different LEO constellation configurations perform in positioning accuracy and interference robustness when operating alone or jointly with GNSS.
Using semi-analytical modeling and 192,000 Monte Carlo simulations, the team evaluated 400 users across European regions in five outdoor scenarios. Key variables included carrier bands (1.5/5/10 GHz), effective isotropic radiated power (EIRP) levels and constellation geometry design.
The team simulated multiple standalone and hybrid constellation architectures, analysing carrier-to-noise ratio (C/N0), geometric dilution of precision (GDOP), position dilution of precision (PDOP) and lower bound 3D accuracy.
Results indicate that an EIRP of 50 dBm is sufficient for high-quality outdoor positioning when operating in L- and C-bands. While 10 GHz platforms require higher power to compensate for path loss, hybrid LEO + GNSS modes show markedly improved stability and reliability.
Multi-shell constellations such as Çelikbilek-1 and Marchionne-2 delivered a favorable balance between satellite count and global geometry, outperforming single-shell layouts. In harsh urban canyon conditions, GNSS accuracy degraded up to seven-fold, whereas LEO-PNT maintained stable ranging performance with limited loss.
Interference resistance also improved. Stronger LEO signal power means jammers require far greater intensity to cause equal degradation. Hybrid designs provided the most significant gains. Combinations such as Çelikbilek-1 + GPS/Galileo, or CentiSpace + BeiDou, yielded better PDOP distributions, faster fix availability and broader user coverage.
The authors conclude that LEO systems are not aimed at replacing GNSS, but rather to enhance availability and resilience under signal-challenged environments.
“Our results show that moderate-power LEO constellations can substantially strengthen outdoor positioning without requiring expensive satellite hardware,” the authors noted. “Geometry plays a major role — carefully designed multi-shell constellations achieve strong accuracy even with fewer satellites. As LEO-PNT develops, hybrid integration with GNSS offers the most cost-effective path toward secure, robust PNT solutions. This work provides guidance for future system designers evaluating frequency, transmission power and constellation configuration trade-offs.”
The findings suggest a realistic rollout pathway for resilient satellite navigation. LEO-enhanced PNT could benefit autonomous vehicles, UAV routing, emergency response, precision farming and critical infrastructure monitoring — especially where GNSS falters in interference-dense or high-rise environments.
Lower-power LEO transmission also reduces deployment cost, opening access for commercial operators.
Future work may assess indoor positioning potential, bandwidth expansion, and real-orbit testing to refine simulation assumptions. As global demand for secure PNT grows, the integration of LEO and GNSS could become a cornerstone for next-generation navigation technology.
The Trektor hybrid robot for agriculture, made by the French company SITIA, can work on a variety of crops by changing the width of its wheelbase and can perform many repetitive tasks, such as spraying and hoeing. (Image: SITIA)
Precision agriculture has been around for more than 30 years and now covers the majority of U.S. farmland. It refers to the ability of farmers to observe, measure and respond precisely to the variability of soil and crop characteristics within and between fields by using maps of these characteristics and GNSS navigation. It enables them to reduce inputs of seed, water, fertilizer, pesticides and fuel while increasing outputs. It also enables them to work at night and in the fog and automate many functions at large feed lots.
For precision agriculture, GNSS integrity can mean the difference between, say, a robot protecting a vineyard by weeding and spraying pesticides or damaging it by straying onto the vines.
Autonomous Tractors, Mowers, and Feed Monitors
SITIA, a French company, has developed an autonomous tractor that is used by, among others, an organic vineyard in France’s Loire valley to tirelessly weed the narrow rows between the grape vines — compensating for the movement of young workers to cities. Thanks to the high accuracy and integrity of the Septentrio GNSS heading receiver inside, the autonomous tractor has decreased the damage to the vineyards by more than an order of magnitude compared to the traditional work done by a farmer with a manual tractor.
Renu Robotics, based in San Antonio, Texas, makes a robot for vegetation management, called Renubot. It uses machine learning, a form of artificial intelligence, to plan its route, optimize its energy consumption, perform self-diagnostics, collect environmental data and assess the topography that it traverses.
Navigation is based on a stored map of paths, a Septentrio RTK GPS receiver and sensors to avoid obstacles. A radio link enables the Renubot to communicate with a control center, for reporting and updates. When the Renubot returns to its recharge pod, it charges its lithium battery and performs updates and downloads.
Manabotix Pty. Ltd., an Australian company, has developed an automated system to monitor cattle in large feedlots, using GNSS, lidar scanning and other vision or perception technologies and artificial intelligence. This has greatly improved the accuracy and consistency of feedlot volume estimates, which for the previous 150 years had been the responsibility of a select few employees, who would visually gauge the amount of feed in concrete troughs. This visual inspection by humans was inherently imprecise, subjective, and inconsistent, often causing animals to eat too much or too little one day and get off their optimal growth curve or even become ill. Manabotix’s solution consists of a Septentrio AsteRx-U GNSS receiver and antenna, a lidar scanner, and an onboard processing platform.
Statistical Analysis
Integrity is a key aspect of all these applications. A part of delivering integrity is a statistical analysis called receiver autonomous integrity monitoring (RAIM), which was developed for such safety-critical applications as aviation or marine navigation. A refinement of RAIM, called RAIM+, takes this analysis to the next level as part of a larger positioning protection package.
For autonomous operation, it can be particularly hazardous to be overly optimistic about GNSS accuracy. This parameter is reported in the form of positioning uncertainty, which is the maximum possible error on the calculated position. It is especially necessary in challenging GNSS environments, where the receiver has a direct line of sight to only a limited number of GNSS satellites or where GNSS signals are degraded. RAIM alerts users when their receiver’s uncertainty strays beyond the limits they have chosen for their application.
Users can be deceived by a consistent position or movement — which can be consistently inaccurate. The positioning uncertainty gives them an indication of the extent to which they can rely on their receiver’s positioning accuracy at any given moment. The receiver operator can set an alarm limit, so that the receiver can flag situations when positioning uncertainty becomes too large.
The blue line in Figure 1 shows position uncertainty estimated by a GNSS receiver under favorable conditions, when the view of the sky is unobstructed, and the receiver has a direct line-of-sight to many satellites.
Figure 1. Under good GNSS conditions, the position uncertainty shown by the blue lines is well within the alarm limits, indicating safe operation. The actual position of the receiver should always remain within the blue uncertainty boundaries. (Image: Septentrio)
During favorable conditions, the positioning uncertainty stays well below the alarm limit because the calculated position is almost the same as the robot’s actual position. However, in challenging environments, the truthfulness of positioning uncertainty becomes most critical (see Figure 2).
Figure 2. In challenging environments receivers with high integrity report large positioning uncertainty, flagging possible inaccuracies to the system. If the receiver is too optimistic about its accuracy, the operation becomes hazardous. (Image: Septentrio)
For instance, when the view of the sky is partially obstructed by buildings or foliage, the receiver has access to only a limited number of GNSS satellites, making it harder to calculate accurate position. In such cases the receiver must report a higher positioning uncertainty, so that the system can take adequate action such as switching to lower speeds, staying further away from predefined boundaries, or stopping.
A low integrity receiver may keep reporting an optimistic positioning uncertainty, that stays below the preset alarm limit even when the calculated position is way off from the actual position. The number may look fine, but effectively it becomes a “robot on the loose,” no longer on its planned path with a risk of damaging itself and its surroundings.
Let us look at uncertainty limits in action during a GNSS car test in an urban canyon, where the view of the sky is partially obstructed by houses (see Figure 3). The orange lines are the positioning and its uncertainty boundaries reported by a Septentrio mosaic GNSS module in the car, while the red lines are the positioning and its uncertainty boundaries reported by another popular GNSS receiver. The white line shows the actual position of the car as it drives along the road. The orange uncertainty boundaries of the mosaic receiver are truthful and somewhat wider in this challenging environment, and you can see that the actual position always remains within these boundaries. On the other hand, the red trajectory jumps off course in a certain challenging spot on the road, with the actual position no more within the uncertainty boundaries, which remain too optimistic. In this case the competitor’s receiver gives a false sense of security and the system is unaware of its hazardous operation.
Figure 3. In an urban canyon car test the Septentrio receiver reports truthful position uncertainty. A competitor receiver seems to be more accurate, while the actual position is not even within its reported uncertainty boundaries. (Image: Septentrio)
If the receiver depicted by the red line provided navigational information for an ADAS automotive system, for example, this could mislead the system into thinking that the car switched lanes. If the system then attempted to correct the trajectory by switching back to the “correct lane” this would result in taking the car off course and potentially hitting the sidewalk or even another car.
RAIM vs RAIM+
The underlying mechanism behind truthful positioning uncertainty reporting is RAIM, which ensures a truthful positioning calculation based on statistical analysis and exclusion of any outlier satellites or signals. Septentrio receivers are designed for high integrity and take RAIM to the next level with RAIM+, guaranteeing truthfulness of positioning with a high degree of confidence.
In Septentrio receivers RAIM+ is a component of a larger receiver protection suite called GNSS+ comprising positioning protection on various levels including AIM+ anti-jamming and anti-spoofing, IONO+ resilience to ionospheric scintillations, and APME+ multipath mitigation.
Septentrio has fine-tuned its RAIM+ statistical model with more than 50 terabytes of field data collected over 20 years. It removes satellites and signals which may give errors due to multipath reflection, solar ionospheric activity, jamming and spoofing, while working together with the GNSS+ components mentioned above. Because of this multi-component protection architecture, it achieves a very high level of positioning accuracy and reliability which goes well beyond the standard RAIM. The RAIM+ statistical model is adaptive, highly detailed, and complete, taking advantage of all available GNSS constellations and signals. The full RAIM+ functionality is also available in Septentrio’s GNSS/INS receiver line. User controlled parameters allow it to be tuned to specific requirements.
The diagram in Figure 4 shows RAIM+ in action during a jamming and spoofing attack on a Septentrio GNSS receiver. While AIM+ removes the effects of GNSS jamming, both AIM+ and RAIM+ work together to block the spoofing attack. Satellites with high distance errors, shown on the middle graph, are removed by RAIM+ since they do not conform to the expected satellite distance.
Figure 4. In this scenario jamming gives satellite distance errors but is countered by AIM+ technology. During spoofing AIM+ eliminates some of the spoofed satellites, while other satellites that have wrong distances are dismissed by RAIM+ algorithms. (Image: Septentrio)
This example shows that even in the case of jamming and spoofing, Septentrio’s high integrity receiver technology delivers truthful and reliable positioning on which any autonomous system can count.
GNSS Design Around Reliability
GNSS receivers designed to be reliable strive for high integrity in both reporting of the positioning uncertainty as well as in RAIM+ advanced statistical modelling. This ensures that these receivers provide truthful and timely warning messages and are resilient in various challenging environments. Other technologies such as inertial navigation system (INS) can also be coupled to the GNSS receiver to extend positioning availability even during short GNSS outages. Quality indicators for satellite signals, CPU status, base-station quality and overall quality allow monitoring of positioning reliability at any given time. High-integrity GNSS receivers provide truthful positioning in autonomous machines such as the SITIA weeding tractor. They are also crucial components in safety-critical applications, assured PNT and any other application where accuracy and reliability matters.
GNSS integrity is key to precision agriculture. (Image: fotokostic/iStock/Getty Images Plus/Getty Images)
In the world of global navigation satellite systems (GNSS), there are five key watchwords: accuracy, integrity, availability, continuity and coverage. While all five of those parameters are very important, their priority order depends on the application.
Accuracy: how well a measured or estimated position or time conforms with its true value. If you are a surveyor, accuracy and integrity are your biggest concerns. Accuracy is not to be confused with precision, which is a measure of exactness. It can refer to the number of significant digits reported for a measurement, the rigor of the measuring process, or the agreement among repeated measurements. For example, for locational error a measurement of 9.725m is more precise than 9.7m but may not be a more accurate measure of the error. A measuring instrument is precise if it repeatedly gives similar measurements, regardless of whether these are actually accurate. They could all be off by, say, 5m, due to some bias in the measurement process. In short, precise data may be inaccurate and accurate data may be imprecise.
Integrity: how much the information supplied by the system can be trusted to be correct. This requires the system to provide timely warnings to the user when the equipment is unreliable for navigation purposes—due to obstructions, jamming, multipath or any other event that degrades accuracy.
Availability: the percentage of time that a signal is available to the user. For location-based services, this and coverage are probably the most important parameters.
Continuity: the ability of the total navigation system to continue to perform its function during the intended operation. Continuity is critical whenever reliance on a particular system is high. For a pilot during an instrument approach procedure, continuity and integrity are vital.
Coverage: the area over which a signal is required. For farmers, it is their fields, for ships, the world’s oceans.
Surveying is both an ancient profession and one of today’s most technologically advanced. Surveyors are among the first on the site of a new construction project, staking out its corners and boundaries, and mapping elevation contours, as well as among the last, surveying the project “as built.” This is particularly important for features that will no longer be visible once the project is complete, such as underground utilities.
While many surveyors work in quiet, uncrowded environments — such as surveying the boundaries of farm fields — those who work on large construction projects operate among the hustle and bustle of bricklayers, carpenters, electricians, plumbers and other tradespeople, as well as cranes, backhoes and other heavy machines. This chaotic environment means that in addition to accuracy and efficiency, surveyors also are concerned with safety.
In the following cover story, a Minnesota-based construction company describes a new system it developed for surveying and mapping underground utilities. Also, professional surveyor Gavin Schrock discusses the benefits of a flexible approach to GNSS rover accuracy and of adding scanning capabilities to robotic total stations.
In recent years, the architecture, engineering and construction (AEC) industry has benefited greatly from growing GNSS accuracy, smaller laser scanners, UAVs, and more efficient management, collaboration and visualization software. We asked five companies operating in this space to address three questions:
What are the key challenges of surveying for the AEC industry today, compared with traditional boundary surveying and other types of surveying?
Which of your products are particularly relevant for this kind of surveying?
What was a recent AEC surveying success story?
In the following articles, five companies briefly describe their experience with the AEC industry:
A PNT expert suggested that my piece titled “Opposite and Complementary: eLoran is part of the solution to GNSS vulnerability” in our November 2021 issue could be augmented with information not currently available on the proposed eLoran capability. This expert also questioned my statement that eLoran “does not have any common failure modes with GNSS” and pointed to potential common threats such as from cyberattacks, physical attacks, and space weather.
Matteo Luccio
I welcome such feedback on the contents of these pages — and agree that in this case some hard questions are warranted. So, in the interest of further exploring the use of eLoran, I pose some questions, hoping that its advocates will provide answers. I know that at least some of them will not shy away from this challenge.
Please note that I wish to keep the discussion on positioning, not the easier question of timing, because that was the primary focus of my article. I also wish to address long-term outages (weeks or months), which would have a greater impact on the United States.
Some of these questions have been addressed, at least in part, in various studies and proposals, most of them now more than a decade old. So, it would be helpful to update those answers and consolidate them in the pages of this magazine.
1. Accuracy specifics. While my November article stated that eLoran would have a two-dimensional accuracy of “better than 20 meters, and in many cases, better than 10 meters,” is that RMS, 95%, or some other statistic?
2. Performance standard. GPS provides a commitment to users in a published performance standard. What specific measures of positioning accuracy, integrity and continuity would you recommend the proposed eLoran system be committed to provide (using the architecture described in the answer to Question 6)?
3. Coverage. Would you recommend this eLoran positioning performance hold for the entire United States (including Alaska, Hawaii, Puerto Rico and other territories), only for the “lower 48” states, or only parts of these 48 states?
4. Current users. By number of users, the predominant common current civil uses of GNSS for positioning are consumer devices (mostly cellphones). By contribution to the U.S. economy, the predominant uses are high-precision applications. For what fraction of these uses would eLoran positioning be adequate? Could an eLoran receiver and antenna fit in today’s consumer devices?
5. Future uses. Emerging civil uses of GPS for positioning include autonomous ground and air vehicles, navigation to space and in space, and lane-accurate car navigation. Which of these could be served by eLoran?
6. Architecture. To maintain accuracy during a prolonged GPS outage, eLoran would require reference stations to calibrate time-varying propagation errors, as well as a certain number of transmitters for good nationwide geometry and for redundancy, ensuring service even if a transmitter is attacked or is taken off-line for maintenance. What architecture would you recommend to achieve this?
7. Infrastructure cost. What would be the cost of installing the required transmitters, power supplies, reference stations, communication links and control system for the architecture described in the answer to Question 6? Can you reference a recent and independent estimate? To a ballpark figure, what cost fixed-price contract would you accept to implement it? Similarly, what would be the annual costs for operating and maintaining this infrastructure?
8. Impact. eLoran transmitters are large and high-power. Providing positioning across the United States could require building some of them from scratch or significantly reconstructing old Loran sites. What issues — such as environmental, aviation safety and security — would this raise, and how would you recommend they be addressed?
9. Receivers. Assuming all the above were achieved, it would accomplish nothing unless eLoran receivers were widely purchased, installed and used. How much would that cost? Who would pay? Should we assume that “if we build it, they will come”?
10. Alternatives. Given the widespread development of other positioning technologies over the past decade, much has changed since the earlier recommendations for eLoran. How do we know that eLoran is the right investment — or even a needed part of the solution or needed system in a system of systems — for the future of U.S. PNT?
Common threats to GNSS and eLoran could include the following:
1. Cyber attacks. Given that GPS’s OCX is said to be the most cybersecure system built by the U.S. Department of Defense, how would eLoran’s control system be even more cybersecure than OCX, to avoid a common cyber-vulnerability?
2. Physical attacks. Given concerns about possible physical attacks on GPS satellites, which move at multiple km/sec 20,000 km from Earth, would it not be easier to physically attack eLoran transmitters, which are stationary, terrestrial, in remote locations, and hundreds of feet tall and require massive power sources?
3. Space weather. GPS is potentially vulnerable to severe space weather that could damage satellites or temporarily hinder signal propagation from space to Earth. However, severe space weather could also damage the power grid upon which megawatt eLoran transmitters rely. How would eLoran service be protected from the effects of severe space weather, such as a Carrington Event?
Send me your thoughts at the e-mail address below, with “eLoran” in the subject line.
Swift Navigation has been named Fleet Management Technology Company of the Year in the second annual AutoTech Breakthrough Awards conducted by AutoTech Breakthrough.
AutoTech Breakthrough is a market intelligence organization that recognizes the top companies, technologies and products in the global automotive and transportation technology markets.
Swift offers a highly-accurate, highly-reliable precise positioning solution that improves the operational efficiency of commercial transport, long-haul trucking and last-mile delivery, whether human-driven or autonomous. Swift’s fleet management precise positioning solution is comprised of the Skylark precise positioning service—delivering continent-wide, cloud-based corrections service — and the receiver-agnostic Starling positioning engine, which works with a variety of automotive-grade GNSS chipsets and inertial sensors, making centimeter-level GNSS accuracy a possibility without the cost of all new equipment.
Swift’s precise positioning solution delivers improved GNSS accuracy to make it easier to enable key fleet management capabilities such as lane-level analytics, route optimization and accurate traffic flow analytics to improve operational efficiency.
New atomic clock technology will improve GNSS location accuracy, as well as addressing the scalability of other quantum technologies being developed
Nanofabrication experts Kelvin Nanotechnology have teamed up with product design specialist Wideblue, the University of Strathclyde and the University of Birmingham on a UK Research and Innovation (UKRI) project funded by the Industrial Strategy Challenge Fund to develop innovative techniques in the miniaturisation of optical atomic clocks.
The new clock technology will help improve GNSS location accuracy, as well as addressing the scalability of other quantum technologies being developed by the academic partners.
“Small, low cost atomic clocks will be essential as we develop a resilient position, navigation and timing (PNT) infrastructure to support our financial, power distribution and communications services,” said Roger McKinlay, challenge director – Quantum Technologies at UKRI.
Cold atomic samples have led to profound advancements in precision metrology by measuring the frequency separation of discrete atomic energy levels. These atomic clocks are the ultimate timekeepers, with the state-of-the-art instruments providing a timing accuracy that it would neither gain nor lose a second in over 30 million years.
Because of the high level of accuracy in these instruments, atomic clocks are used to coordinate systems that require extreme precision, such as GNSS. Each satellite network contains multiple atomic clocks that contribute precision timing data, which is decoded to provide location data by effectively synchronizing each receivers’ atomic clocks with those of the satellite.
“The project is a feasibility study which aims to facilitate the miniaturization of state-of-the-art atomic clocks.” said Russell Overend, managing director of Wideblue. “To achieve such high timing resolution, the atomic clock makes use of ultra-narrow transitions in strontium atoms, providing orders of magnitude better performance than their rubidium counterparts due to narrower atomic features. In simple terms, the narrower the atomic transition the more accurate the atomic clock.
At Strathclyde, cold atom clock experiments are aided by expertise in grating magneto-optical traps (gMOTs), illustrated here. (Image: Aidan Arnold, University of Strathclyde)
An important factor in cold atomic clock technology is grating magneto-optical traps (gMOTs). With gMOTs, diffraction gratings split and steer an incoming beam into a tripod of diffracted beams, allowing trapping in the four-beam overlap volume.
Wideblue will develop the optical system that will deliver the laser light onto the gMOT chip. Kelvin Nanotechnology will manufacture the gMOT and compact collimation optics designed by Wideblue. The University of Strathclyde will design the gMOT chip, and the University of Birmingham will perform the testing of the prototype optical system.
“Atomic clocks are an integral component in modern technology and impact our daily routines from computing and financial transactions to the navigation systems we use in our phones and cars,” said James McGilligan, Kelvin Nanotechnology, “As state-of-the-art atomic clocks push new boundaries in precision measurement, we face a new challenge of bringing this complex and large physical apparatus into a compact and user-friendly system where we can make the largest societal and economic impact.
“Our current collaboration with Wideblue and our academic partners aims to address the scalability of one such atomic clock by reducing the optical constraints into scalable micro-fabricated components as a critical step to bringing laboratory performance out into real world applications,” McGilligan said.
“With support from the Quantum Technologies Challenge in UKRI — part of the UK National Quantum Technologies Programme — we are ensuring that the UK economy and society will benefit from the next generation of quantum devices and be quantum ready,” McKinlay said.
The number of GNSS constellations, satellites and signals is constantly growing. The threats to GNSS — from unintentional radio frequency interference (RFI), jamming, spoofing, multipath… and Federal Communications Commission rulings — are increasing, as are the public’s expectations of GNSS accuracy.
All these factors contribute to the need for ever more powerful and advanced simulators that can realistically simulate a wide range of optimal and suboptimal environments. That is why simulators are a rapidly growing sector of the GNSS industry.
At present, the main defense against jamming are continuous radiation pattern antennas (CRPA). Therefore, it is essential that simulators be able to accurately reproduce signals from CRPAs. They are even more useful when they can generate M-code (MNSA) signals, which not all simulators do.
Additionally, the development of autonomous vehicles requires engineers to simulate driving millions of miles, under a variety of environmental and traffic circumstances. To accomplish this in a reasonable amount of time requires them to run simulations faster than in real time, or run many simulations in parallel.
Finally, there is an increasing need to simulate alternative positioning, navigation and timing (PNT) signals being developed as supplements to and substitutes for GNSS signals in circumstances that make the latter unavailable or unreliable.
These are some of the challenges facing manufacturers of GNSS simulators. What follows are their brief descriptions of the approaches they are taking and the innovations they are introducing.
What is your most recent innovation?
Our latest simulator innovations contain wave-front generation signal technology, which allows you to generate GNSS and interference signals that represent the received signals for each antenna element in a phased array antenna manifold, usually referred to as a controlled radiation pattern antenna (CRPA). Our modular design approach enables users to simulate IMU data commensurate with the wave-front signals for a complete coherent GNSS/IMU simulation that is ideal for stimulating receivers that contain CRPA and IMU capabilities. Our simulators also contain proprietary synchronization technology that allows users to synchronize multiple systems to produce a “wave-front” of GNSS and IMU signals for multiple vehicles, or even an entire fleet.
Photo: CAST Navigation
What is your approach to jamming and spoofing?
CAST Navigations’ family of GNSS simulators are capable of realistically simulating a wide range of suboptimal conditions—such as jamming/spoofing, multipath, RF interference and satellite constellation perturbations—for virtually any commercial or military environment. Our interference signals or “jammers” can be located at any terrestrial location and can be static or dynamic in nature. A distinguishing feature of CAST Navigations’ simulation systems is that our interference signals are phase-controlled and coherent, allowing for proper phase transmission of each signal type for each receiving antenna element. You can also add an INS capability to any of our systems. These types of systems are perfect for testing GNSS and GNSS/INS types of navigation equipment.
What’s coming by 2023?
One of the key trends is the ability to generate M-code (MNSA) signals. Jamming and spoofing are becoming more prevalent, not just to the military but also to consumers. Every day, the military, as well as people like you and me, are starting to encounter more instances of interference that can deny GNSS equipment and even phones the ability to track some GNSS satellites or that transmit incorrect GNSS data, causing receivers to display incorrect position solutions. So, our focus is on products and capabilities that enable our customers to simulate those types of environments and help them to mitigate those kinds of events.
Orolia
Lisa Perdue Product Manager
What is your most recent innovation?
At Orolia we continue to evolve our innovative software-defined simulator approach. Our most recent innovation is our advanced spoofing option. We have taken our ability to define multiple jamming transmitters, each with their own trajectory and antenna pattern, and added the ability for the transmitters to send spoofing signals as well. By utilizing our capability to run multiple simulations on a single system, we give the user the ability to control every parameter of the generated spoofing constellation(s). The system automatically calculates the signal time of flight and the propagation loss, making this advanced capability powerful and easy to use.
What is your approach to jamming and spoofing?
Simulation of threat environments is a critical component of GNSS receiver testing. As awareness of the impact that jamming and spoofing can have on a GNSS-based system rises, so does the need to test. That is why we have implemented advanced jamming and spoofing options into our Skydel simulator’s core engine. Replication of degraded environments with threats ranging from one to hundreds is possible using the same hardware and software used for generating GNSS signals. No third-party hardware or software is required for complete testing against jamming and spoofing because we feel that this capability should be part of the core system, not an afterthought.
Photo: Orolia
What’s coming by 2023?
In the coming years, we expect to see more requirements for simulation of alternative positioning, navigation, and timing (PNT) signals. As governments and organizations continue to investigate alternate technologies, it will become necessary to simulate low Earth orbit (LEO) PNT, ground-based transmitters, and other signals being considered.
Another growing trend is the adoption of controlled reception pattern antennas (CRPAs) for their anti-jam capabilities. These anti-jam antenna systems can only be tested by specialized simulation systems, so we can imagine these simulation systems being commercialized for a broader market around 2023.
Racelogic
Julian Thomas Managing Director
What is your most recent innovation?
Recognizing the need of our customers to test their products with a simple solution that uses the latest GNSS signals, we have updated our SatGen software to create accurate simulations using all satellite data currently being transmitted across the various constellations. We have also optimized the performance of SatGen so that a standard desktop PC can be used to simulate these signals in real time. Also, the simulation can now be driven using an external NMEA stream, allowing full remote control of the trajectory.
What is your approach to jamming and spoofing?
The LabSat 3 Wideband records and replays all available GNSS signals in high fidelity, allowing jamming and spoofing signals to be reproduced accurately on the test bench.
Photo: Racelogic
What’s coming by 2023?
With so many employees now working from home due to COVID-19, the pressing concern for many companies developing GNSS technology is how to provide employees with suitable equipment that is required for them to carry out their jobs efficiently away from the office. Usually these employees would utilize the shared resources of a well-equipped office, with experts on hand to help, but working from home has made access to these devices challenging. Due to LabSat 3’s small size, low cost and ease of use, we have seen a significant increase in sales to companies furnishing their employees with a suitable method of testing their GNSS devices while working from home.
With the advent of a new breed of high-performance, low-cost GNSS receiver, many new applications are being developed in new and exciting sectors, utilizing a level of accuracy previously considered too expensive to be a commercial proposition. The number of GNSS engines will therefore increase rapidly in the marketplace, with a corresponding increase in demand for cost-effective signal simulation for test and development.
Rohde & Schwarz
Markus Irsigler Product Manager Signal Generators, Power Meters
What is your most recent innovation?
We further improved multi-frequency, multi-constellation simulation capabilities in our high-end segment. The GNSS high-end simulator R&S SMW200A provides signals for all GNSS frequency bands on a single RF output. A second internal RF path can be used for advanced interferer simulation, testing the receiver’s resilience to spoofing or to address dual-antenna scenarios. This keeps setups simple and compact. When more than two RF paths are required, two or more R&S SMW200A can be operated together in a master/slave configuration. Such setups are required for multi-antenna receiver test applications where the signals’ relative carrier phases are analyzed, like CRPA or attitude determination tests. Our new RF ports alignment software automates alignment of the GNSS signals and guarantees correct amplitude, time and phase relations at the RF inputs of the device under test. We also increased the maximum channel count to more than 600 channels to improve testing of multi-constellation, multi-frequency receivers against multipath, jamming and spoofing.
What is your approach to jamming and spoofing?
Besides our recent innovations, Rohde & Schwarz plans to provide new interference simulation capabilities within the GNSS simulator. This new feature will allow the user to replay recorded jammer signals as well as user-defined waveforms. The R&S Pulse Sequencer software helps with the definition of most complex interferer scenarios.
Photo: Rohde & Schwarz
What’s coming by 2023?
Developments in the field of advanced driver-assistance systems (ADAS) aiming for fully autonomous vehicles raise new challenges for reliable PNT solutions. Simulation of interference and jamming scenarios will hence become important in the automotive market. Antenna arrays have proven suitable to counteract RF interference (RFI) by incorporating spatial-processing techniques and might therefore find greater entry into the automotive market. Test solutions must address requirements for simulating all kinds of intentional and unintentional RFI for multi-constellation, multi-frequency and multi-antenna receivers. Apart from simulating GNSS and interference sources, test solutions for autonomous driving will require several other techniques and signals to be applied or simulated, such as RTK/PPP or outputs from other vehicle sensors to perform sensor fusion.
Spirent Federal Systems
Jeff Martin Vice President, Sales
What is your most recent innovation?
Launched in 2018, SimMNSA became the first MNSA simulator to achieve GPS Directorate security approval. The software enables users to simulate true MNSA M-code with real-time code and message generation, removing the constraints imposed by simulator data sets (SDS). SimMNSA v2.0 does even more. It is able to broadcast nominal M-code conditions and recreate SDS-defined events. It incorporates an advanced editor to edit military navigation (MNAV) content, allows users to craft and define scenarios, and much more.
What is your approach to jamming and spoofing?
Spirent offers numerous capabilities for emulating GNSS signals in the presence of interference and spoofing attacks. Our solutions provide accurate, repeatable and quantifiable signals, enabling customers to conduct accurate tests with trusted results. We can test against internally generated interference enabling multiple “fields” of jammers with various interference types; hundreds of interference signals using external IQ blended with simulator-generated GNSS, and Blue Force Electronic Attack jamming waveforms for testing MGUE devices operating in GPS-denied environments. Spoofing capabilities include signal, navigation data and cyber-level attacks via manipulation of up to 12 copies of each primary GNSS constellation, each fully editable; intuitive spoof attack generation via Spirent’s SimSAFE software option — which also allows live sky synchronization/spoofing, and more.
Photo: Spirent
What’s coming by 2023?
Threats to reliable and accurate GNSS navigation and timing are developing rapidly. Fortunately, innovative solutions for resilient PNT are in development and will continue to challenge the industry for years to come. The ability to simulate these threats and the mitigation techniques to overcome them is changing the landscape for the simulator industry. It’s more important than ever to have up-to-date test tools. Robust signals along with frequency and constellation diversity will continue to drive the market in addition to GNSS backup systems, or AltNav. The FCC has certainly presented the GNSS industry with an immense challenge.
Syntony GNSS
Sylvain Daubas Simulator Activity Manager
What is your most recent innovation?
Yesterday, GPS systems had to “work.” Today, they must work fine. This is the difference, and all equipment vendors have realized this. It is no longer acceptable to have 200 meters or more of error in an urban environment. Because of the extreme complexity of the electromagnetic situation in the GNSS spectrum, making a reliable and precise location system requires more and more powerful and advanced simulators. This is why the GNSS simulator market is booming.
Among the many new features implemented in Syntony’s GNSS simulator this year, two stand out.
First, 1000-Hz hardware-in-the-loop now allows an accurate simulation for high-dynamic receivers (up to more than 100 Gs!), with zero artifact and zero-effective latency. This is the ultimate in trajectory management.
Second, signal computing capacity has made a significant leap forward due to hardware and software optimizations. Constellator can now simultaneously generate up to 660 L1 C/A-equivalent signals. And this level of performance can be unlocked remotely, without a hardware update.
Photo: Syntony GNSS
What is your approach to jamming and spoofing?
Simulating a GNSS environment with a set of jamming or spoofing signal sources today is the standard. But what about a simulation of an extremely complex urban scene with 50 or 100 jamming/spoofing sources? The only reasonable solution to implement this would be a massive parallel software-defined radio (SDR)-based simulator solution. This is what Syntony can and will do, thanks to its full software GNSS simulator architecture, which can be distributed on a server farm.
What’s coming by 2023?
A revolution is arriving: the possibility of generating a full GNSS simulation including many hundreds of satellites and signals, in real time and in pure software. This is now possible, and Syntony has demonstrated it with the Constellator. This will change the simulation world. First of all, Moore’s law will bring significant improvements to this domain year after year. More importantly, new systems and services will be possible: massive parallel scenario simulation including jamming and spoofing, floating simulator licenses, software as a service, etc. In this trend, playback machines will be needed, and obviously a strong internet connection will be necessary to download hundreds of gigabytes of I/Q files overnight.
Feature image: Samuel King Jr./United States Air Force
Leica Geosystems has introduced the Leica Zeno GG04 smart antenna, enabling a flexible solution to improve mobile devices’ GNSS accuracy with real-time kinematic (RTK) and precise point positioning (PPP).
Paired with the Zeno GG04, any Zeno or third-party mobile device with Android or Windows OS can now collect highly precise positioning data with Leica Geosystems’ GNSS technology and 555-channel tracking performance. With PPP, users can collect data in areas without cellular coverage. The bring-your-own-device (BYOD) functionality enables any smart device to collect survey-grade data, delivering centimeter results.
“We’re excited to hear about the new Zeno Connect for Android. Being able to connect any Android device to the new GG04 antenna and use it for field data capture is a real game-changer,” said Zenny Chareas, project manager at PeopleGIS, a firm that builds web-based database applications for field collection currently using the Leica Zeno GG03. “Our clients have been eagerly anticipating this type of functionality, and it’s pretty cool that we now have a solution for them.”
With the Zeno Connect app, any third-party app is compatible with the Zeno GG04 smart antenna. The Zeno Mobile, Zeno Connect or Esri’s Collector for ArcGIS apps provide an easy and familiar platform for non-surveying professionals to collect and analyze data. Organizations can integrate and enrich data in real time from different sources to collect all details of any project from anywhere in the world, regardless of how remote.
“Wherever users are working, despite, how rough the environment, the Zeno GG04 ensures all needed data is easily and accurately collected,” said Alexander Fischer, Leica Geosystems Zeno product manager. “The flexibility offered by turning our most common devices into precise instruments increases access to the geopositioning world, and this is certainly an exciting advancement to share technology and information with new segments.”
Sponsored by: Hemisphere Broadcast Date: Thursday, June 20, 2013 Moderator:Eric Gakstatter, Survey Scene Newsletter Editor Speakers: Kevin Kelly, Geodesist, ESRI, Inc.; Craig Greenwald, Technical Director, GeoMobile Innovations; Michael L. Dennis, RLS, PE, Geodesist, NOAA
A look at the challenge of dealing with horizontal datums in your GIS. We are moving into a new era in dealing with datum transformations. Geodata 2.0 is coming, and it can create big headaches when attempting to combine disparate geospatial databases. Sensors such as GPS receivers, remote sensing imagery, and 3D scanning provide much more accurate data, setting up a collision with outdated and mismatched legacy horizontal datums.
True to its word origins, accuracy demands careful and thoughtful work. This article provides a close look at the differences between the precision and accuracy of GPS-determined positions, and should alleviate the confusion between the terms — making abuse of the truth perhaps less likely in the business of GPS positioning.
INNOVATION INSIGHTS by Richard Langley
JACQUES-BÉNIGNE BOSSUET, the 17th century French bishop and pulpit orator, once said “Every error is truth abused.” He was referring to man’s foibles, of course, but this statement is much more general and equally well applies to measurements of all kinds. As I am fond of telling the students in my introduction to adjustment calculus course, there is no such thing as a perfect measurement. All measurements contain errors. To extract the most useful amount of information from the measurements, the errors must be properly analyzed.
Errors can be broadly grouped into two major categories: biases, which are systematic and which can be modeled in an equation describing the measurements, thereby removing or significantly reducing their effect; and noise or random error, each value of which cannot be modeled but whose statistical properties can be used to optimize the analysis results.
Take GPS carrier-phase measurements, for example. It is a standard approach to collect measurements at a reference station and a target station and to form the double differences of the measurements between pairs of satellites and the pair of receivers. By so doing, the biases in the modeled measurements that are common to both receivers, such as residual satellite clock error, are canceled or significantly reduced. However, the random error in the measurements due to receiver thermal noise and the quasi-random effect of multipath cannot be differenced away. If we estimate the coordinates of the target receiver at each epoch of the measurements, how far will they be from the true coordinates?
That depends on how well the biases were removed and the effects of random error. By comparing the results from many epochs of data, we might see that the coordinate values agree amongst themselves quite closely; they have high precision. But, due to some remaining bias, they are offset from the true value; their accuracy is low. Two different but complementary measures for assessing the quality of the results.
In this month’s column, we will examine the differences between the precision and accuracy of GPS-determined positions and, armed with a better understanding of these often confused terms, perhaps be less likely to abuse the truth in the business of GPS positioning.
“Innovation” features discussions about advances in GPS technology, its applications, and the fundamentals of GPS positioning. The column is coordinated by Richard Langley, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
For many, Global Positioning System (GPS) measurement errors are a mystery. The standard literature rarely does justice to the complexity of the subject. A basic premise of this article is that despite this, most practical techniques to evaluate differential GPS measurement errors can be learned without great difficulty, and without the use of advanced mathematics. Modern statistics, a basic signal-processing framework, and the careful use of language allow these disruptive errors to be easily measured, categorized, and discussed.
The tools that we use today were developed over the last 350 years as mathematicians struggled to combine measurements and to quantify error, and to generally understand the natural patterns. A distinguished group of scientists carried out this work, including Adrien-Marie Legendre, Abraham de Moivre, and Carl Friedrich Gauss. These luminaries developed potent techniques to answer numerous and difficult questions about measurements.
We use two special terms to describe systems and methods that measure or estimate error. These terms are precision and accuracy. They are terms used to describe the relationship between measurements, and to underlying truth. Unfortunately, these two terms are often used loosely (or worse used interchangeably), in spite of their specific definitions. Adding to the confusion, accuracy is only properly understood when divided into its two natural components: internal accuracy and external accuracy.
GPS measurements are like many other signals in that with enough samples the probability distribution for each of the three components is typically bell-shaped, allowing us to use a particularly powerful error model. This bell-shaped distribution is often called a Gaussian distribution (after Carl Friedrich Gauss, the great German mathematician) or a normal distribution. Once enough GPS signal is accumulated, a normal distribution forms. Then, potent tools like Gauss’s normal curve error model and the associated square-root law can be brought to bear to estimate the measurement error.
An interesting aspect of GPS, however, is that over short periods of time, data are not normally distributed. This is of great importance because many applications are based upon small datasets. This results in a fundamental division in terms of how measurement error is evaluated. For short periods of time, the gain from averaging is difficult to quantify, and it may or may not improve accuracy. For longer periods of time the gain from averaging is significant, a normal distribution forms, and the square-root law is used to estimate the gain. The absence of a Gaussian distribution in these datasets (1 hour or less) is one source of the confusion surrounding measurement error. Another source of confusion is the richly nuanced concept of accuracy. By closely looking at each of these, a clear picture emerges about how to effectively analyze and describe differential GPS measurement error.
The GPS Signal
It is helpful to consider consecutive differential GPS measurements as a signal, and thus from the vantage of signal processing. Here, we use the term measurement to refer to position solutions rather than the raw carrier-phase and pseudorange measurements a receiver makes. Sequential position measurements from a GPS system are discrete signals, the result of quantization, transformation, and other processing of the code and carrier data into more meaningful digital output. In comparison, a continuous signal is usually analog based and assumes a continuous range of values, like a DC voltage. A signal is a way of describing how one value is related to another.
Figure 1 shows a time series consisting of a discrete signal from a typical GPS dataset (height component). These data are based on processing carrier-phase data from a pair of GPS receivers, in double-difference mode, holding the position of one fixed while estimating that of the other. The vertical axis is often called the dependent variable and can be assigned many labels. Here it is labeled GPS height. The horizontal axis is typically called the independent variable, or the domain. This axis could be labeled either time or sample number, depending on how we want this variable to be represented. Here it is labeled sample number. The data in Figure 1 are in the time domain because each GPS measurement was sampled at equal intervals of time (1 second). We’ll refer to a particular data value (height) as xi.
Figure 1. A 10-minute sample of GPS height data.
Ten minutes of GPS data are displayed in Figure 1. These data are the first 600 measurements from a larger 96-hour dataset that forms the basis of this paper. The mean (or average) is the first number to calculate in any error-assessment work. The mean is indicated by . There is nothing fancy in computing the mean; simply add all of the measurements together and divide by the total sample number, or N. Equation 1 is its mathematical form:
[1]
The mean for these data is 474.2927 meters, and gives us the average value or “center” of the signal. By itself, the mean provides no information on the overall measurement error, so we start our investigation by calculating how far each GPS height determination is located away from the mean, or how the measurements spread or disperse away from the center. In mathematical form, the expression denotes how far the ith sample differs from the mean.
As an example, the first sample deviates by 0.0038 meters (note that we always take the absolute value). The average deviation (or average error) is found by simply summing the deviations of all of the samples and dividing by N. The average deviation quantifies the spreading of the data away from the mean, and is a way of calculating precision. When the average deviation is small, we say the data are precise. For these data, the average deviation is 0.0044 meters.
For most GPS error studies, however, the average deviation is not used. Instead, we use the standard deviation where the averaging is done with power rather than amplitude. Each deviation from the mean, , is squared, , before taking the average. Then the square root is taken to adjust for the initial squaring. Equation 2 is the mathematical form of the standard deviation (SD):
[2]
The standard deviation for the data in Figure 1 is 0.0052 meters.
But note that these data have a changing mean (as indicated by the slowly varying trend). The statistical or random noise remains fairly constant, while the mean varies with time. Signals that change in this manner are called nonstationary. In this 10-minute dataset, the changing mean interferes with the calculation of the standard deviation. The standard deviation of this dataset is inflated to 0.0052 meters by the shifting mean, whereas if we broke the signal into one-minute pieces to compensate, it would be only 0.0026 meters.
To highlight this, Figure 2 is presented as an artificially created (or synthetic) dataset with a stationary mean equal to the first data point in Figure 1, and with the standard deviation set to 0.0026 meters. This figure, with its stable mean and consistent random noise, displays a Gaussian distribution (as we will soon see graphically), and illustrates what our dataset is not.
Figure 2. A 10-minute sample of synthetic data.
Contrasting these two datasets helps us to understand a critical aspect of differential GPS data. Analyzing a one-minute segment of GPS data from Figure 1 would provide a correct estimate of the standard deviation of the higher frequency random component, but would likely provide an incorrect estimate of the mean. This is because of its wandering nature; a priori we do not know which of the 10 one-minute segments is closer to the truth. It is tempting then to think that by calculating the statistics on the full 10 minutes we will conclusively have a better estimate of the mean, but this is not true.
The mean might be moving toward or away from truth over the time period. It is not yet centered over any one value because its distribution is not Gaussian. What’s more, when we calculate the statistics on the full 10 minutes of data, we will distort the standard deviation of the higher frequency random component upwards (from 0.0026 meters to 0.0052 meters).
This situation results in a great deal of confusion with respect to the study of GPS measurement error. When we look at Figures 1 and 2 side by side we see the complication. Figure 2 is a straightforward signal with stationary mean and Gaussian noise. Averaging a consecutive series of data points will improve the accuracy. Figure 1 is composed of a higher frequency random component (shown by the circle), plus a lower frequency non-random component. It is the superimposition of these two that causes the trouble. We cannot reliably calculate the increase in accuracy as we accumulate more data until the non-random component converges to a random process. This results in a very interesting situation; in numerous cases gathering more data can actually move the location parameter (the mean, ) away from truth rather than toward it.
To fully understand the implications of this, consider its effect on estimating accuracy. If the mean is stationary, statistical methods developed by Gauss and others could be used to estimate the measurement error of an average for any set of N samples. For example, the so-called standard error of the average (SE) can be computed by taking the square root of the sample number, multiplying it by the standard deviation, and then dividing by the sample number (a method to provide an estimate of the error for any average that is randomly distributed). Equation 3 is its mathematical form:
[3]
which simplifies to S/√N . This model can only be used if the data have a Gaussian distribution. Clearly this model cannot be used for the data in Figure 1, but can be used for the data in Figure 2. The implications are significant. The data from Figure 1 are not Gaussian because of the nonstationary mean, so we do not know if the gain from 10 minutes of averaging is better or worse than the first measurement. By contrast, the data in Figure 2 are Gaussian, so we know that the average of the series is more accurate than any individual measurement by a factor equal to the square root of the measurements.
By shifting these data into another domain we can see this more clearly. Figure 3 shows the 10 minutes of GPS data from Figure 1 plotted as a histogram or distribution of the number of data values falling within particular ranges of values. We call each range a bin. The histogram shows the frequencies with which given ranges of values occur. Hence it is also known as a frequency distribution. The frequency distribution can be converted to a probability distribution by dividing the bin totals by the total number of data values to give the relative frequency. If the number of observations is increased indefinitely and simultaneously the bin size is made smaller and smaller, the histogram will tend to a smooth continuous curve called a probability distribution or, more technically, a probability density function. A normal probability distribution curve is overlain in Figure 3 for perspective. This curve simultaneously demonstrates what a normal distribution looks like, and serves to graphically display the underlying truth (by showing the correct frequency distribution, mean, and standard deviation). It was generated by calculating the statistics of the 96-hour dataset, then using a random-number generator with adjustable mean and standard deviation (this is an example of internal accuracy, and will be discussed at length in an upcoming section). We can see that our Figure 1 dataset is not Gaussian because it does not have a credible bell shape. By contrast, when we convert the synthetic data from Figure 2 into a frequency distribution, we see the effect of the stationary mean — the data are distributed in a normal fashion because the mean is not wandering.
Figure 3. Frequency distribution of a 10-minute sample of GPS height data.
Recall that all that is needed to use the Gauss model of measurement error is the presence of a random process. Mathematically, the measurement accuracy for the average of the data in Figures 1 and 3 is the overall standard deviation, or 0.0052 meters, because there is no gain per the square-root law. In comparison, the measurement accuracy for the average in Figure 4 is SE = (√ 600•0.0026) / 600 = 0.0001 meters. The standard deviation from the mean is still 0.0026 meters, but the accuracy of the averaged 600 samples is 0.0001 meters. Recall that precision is the spreading away from the mean, whereas accuracy is closeness to truth. When a process is normally distributed, the more data we collect the closer we come to underlying truth. The difference between the two is remarkable. Measurement error can be quickly beaten down when the frequency distribution is normal. This has significant implications for people who collect more than an hour of data, and raises the following question: At what point can we use the standard error model?
Figure 4. Frequency distribution of a 10-minute sample of synthetic data.
Frequency Distribution
In an ideal world, GPS data would display a Gaussian distribution over both short and long time intervals. This is not the case because of the combination of frequencies that we saw earlier (random + non-random). As an aside, this combination is a good example of why power is used rather than amplitude to calculate the deviation from the mean. When two signals combine, the resultant noise is equal to the combined power, and not amplitude.
Interesting things happen as we accumulate more data and continue our analysis of the 96-hour dataset. Earlier we discussed calculating the SD and the mean, and we looked at short intervals of GPS data in the time domain and the frequency-distribution domain. Moving forward, we will continue to look at the data in the frequency-distribution domain because it is far easier to recognize a Gaussian distribution there. The goal is to discover the approximate point at which GPS data behave in a Gaussian fashion as revealed by the appearance of a true bell curve distribution.
Figure 5 shows one minute of GPS data along with the “truth” curve for perspective. This normal curve, as discussed above, was generated using a random number generator with programmable SD and mean variables. The left axis shows the probability distribution for the GPS data, and the right axis shows the probability distribution function for the normal curve. This figure reinforces what we already know: one minute of GPS data are typically not Gaussian (Figure 3 shows the same thing for 10 minutes of data).
Figure 5. Frequency distribution of a 1-minute sample of GPS height data.
Figure 6 shows 1 hour of GPS data. The data in Figure 6 show the beginnings of a clear normal distribution. Note that the mean of the GPS data is still shifted from the mean of the overall dataset. The appearance of a normal distribution at around 1 hour of data indicates that we can begin use of the standard error model, or the Gaussian error model. Recall that this states that the average of the collection of measurements is more accurate that any individual measurement by a factor equal to the square root of the number of measurements, provided the data follow the Gauss model and are normally distributed. For one hour of data, the gain is square root of 1 times the SD divided by N. In effect, no gain. But from this point forward each hour of data provides √N gain. Figure 7 shows 12 hours of data with a gain of √12. By calculating the standard error for the average of 12 hours of data, SE = (√12•0.0069)/12, or 0.0020 meters, we see a clear gain in accuracy. Notice also that at 12 hours the normal curve and the GPS data are close to being one and the same.
Figure 6. Frequency distribution of a 1-hour sample of GPS height data.
Figure 7. Frequency distribution of a 12-hour sample of GPS height data.
Several things are worth pointing out here. The non-stationary mean converts to a Gaussian process after approximately 1 hour. There is nothing magical about this; conversion at some point is a necessary condition for the system to successfully operate. If it did not, the continually wandering mean would render it of little use as a commercial positioning system. Because it is non-stationary over the shorter occupations considered normal for many applications, it is confusing. Collecting more data in some instances can contribute to less accuracy. This situation also creates a gulf between those who collect an hour or two, and those who collect continuously. It is worth emphasizing that the distribution of data under our “truth” curve fills out nicely after 12 hours. This coincides with one pass of the GPS constellation, suggesting (as we already know) that a significant fraction of the wandering mean is affected by the geometrical error between the observer and the space vehicles overhead.
By looking at the 12 one-hour Gaussian distributions that comprise a 12-hour dataset, we see clearly what Francis Galton discovered in the 1800s. A normal mixture of normal distributions is itself normal, as Figure 8 shows. This sounds simple, but in fact it has significant implications. The unity between consecutive 1-hour segments of our dataset is the normal outline, reinforcing the increasing accuracy of the location parameter, , as more and more normal curves are summed together.
Figure 8. (a) Frequency distribution of 12 1-hour samples of GPS height data; (b) the 12 1-hour samples combined.
Internal vs. External Accuracy
Figure 9 shows the relationship between precision and accuracy. The dashed vertical line indicates the mean of the dataset (the inflection point at which the histogram balances). The red arrows bracket the spread of the dataset at 1 standard deviation from the mean (precision), while the black arrows bracket the offset of the mean from truth (accuracy). Notice that the mean ( ) is a location parameter, while the standard deviation (<e
m>s) is a spread parameter. What we do with the mean is accuracy related; what we do with the standard deviation is precision related.
Figure 9. Relationship between precision and accuracy.
Accuracy is the difference between the true value and our best estimate of it. While the definition may be clear, the practice is not. Earlier we discussed two techniques used to calculate precision — the average deviation, and the standard deviation. We also discussed the square-root law that estimates the measurement error of a series of random measurements. As we saw, it was not possible to calculate this until roughly 1 hour of data had been collected. Furthermore, the data were said to be accurate when a good correlation appeared between the overlain curve and the GPS data at 12 hours.
But here is the interesting thing; the truth curve was derived internally. As previously discussed, data were accumulated for 96 hours, and then statistics were calculated on the overall dataset. Then a random number generator with programmable mean and standard deviation was used to generate a perfectly random distribution curve with the same location parameter and spread. This was declared as truth, and then smaller subsets of the same dataset were essentially compared with a perfect version of itself! This is an example of what is called internal accuracy.
By contrast, external accuracy is when a standard, another instrument, or some other reference system is brought to bear to gauge accuracy. A simple example is when a physical standard is used to confirm a length measurement. For instance, a laser measurement of 1 meter might be checked or calibrated against a 1-meter platinum iridium bar that is accepted as a standard. The important point here is that truth does not just appear — it has to be established through an internal or external process.
Accuracy can be evaluated in two ways: by using information internal to the data, and by using information external to the data. The historical development of measurement error is mostly about internal accuracy. Suppose that a set of astronomical measurements is subjected to mathematical analysis, without explicit reference to underlying truth. This is internal accuracy, and was famously expressed by Isaac Newton in Book Three of his Principia: “For all of this it is plain that these observations agree with theory, so far as they agree with one another.”
Internal accuracy constrains and simplifies the problem. It eliminates the need to bring other instruments or systems to bear. It makes the problem manageable by allowing us to use what we already have. Most importantly, it eliminates the need to consider point of view. Because we are not venturing outside of the dataset, it becomes the reference frame. By contrast, when you ponder bringing an external source of accuracy to bear it gets complicated, especially with GPS.
For example, is it sufficient to use one GPS receiver to check the accuracy of another, or should an entirely different instrument be used? Is it suitable to use the Earth-centered, Earth-fixed GPS frame to check itself, or should another frame be used? If we use another frame, should it extend beyond the Earth, or is it sufficient to consider accuracy from an Earth perspective? When we say a GPS measurement is accurate, what we are really saying is that it is accurate with respect to our reference frame. But what if you were an observer located on the Sun? An Earth-centric frame no longer makes sense when the point that you wish to measure is located on a planet that is rotating in an orbit around you. For an observer on the Sun, a Sun-centered, Sun-fixed reference frame would probably make more sense, and would result in easier to understand measurements. But we are not on the Sun, so a reference frame that rotates with the Earth — making fixed points appear static — makes the most sense. The difference between the two is that of perspective, and it can color our perception of accuracy.
Internal accuracy assessments sidestep these complications, but make it difficult to detect systematic errors or biases. Keep in mind that any given GPS measurement can be represented by the following equation: measurement = exact value + bias + random error. The random-error component presents roughly the same problem for both internal and external assessments. The bias however, requires external truth for detection. There is no easy way to detect a constant shift from truth in a dataset by studying only the shifted dataset.
In practice, people generally look for internal consistency, as Newton did. We look for consistency within a continuous dataset, or we collect multiple datasets at different times and then look for consistency between datasets. It is not uncommon to use the method taken in this article: let data accumulate until one is confident that the mean has revealed truth, and then use this for all further analysis. For this approach, accuracy implies how the measurements mathematically “agree with one another.”
All of this shows that accuracy is a very malleable term. Internal accuracy assumes that the process is centered over truth. It is implicitly understood that more measurements will increase the accuracy once the distribution is normal. The standard error is calculated by taking the square root of the sample number, multiplying it by the standard deviation, and then dividing by the sample number. With more samples, the standard error of the average decreases, and we say that the accuracy is increasing. Internal accuracy is a function of the standard deviation and the frequency distribution.
External accuracy derives truth from a source outside the dataset. Accuracy is the offset between this truth and the measurement, and not a function of the standard deviation of the dataset. The concept is simple, but in practice establishing an external standard for GPS can be quite challenging. For counterpoint, consider the convenient relationship between a carpenter and a tape measure. He is in the privileged position of carrying a replica of the truth standard. GPS users have no such tool. It is impossible to bring a surrogate of the GPS system to bear to check a measurement. Fortunately, new global navigation satellite systems are coming on line to help, but a formal analysis of how to externally check GPS accuracy leads one into a morass of difficult questions.
Accuracy is not a fundamental characteristic of a dataset like precision. This is why accuracy lacks a formal mathematical symbol. One needs to look no further than internal accuracy for the proof. For a dataset that is shifted away from truth, or biased, no amount of averaging will improve its accuracy. Because it is possible to be unaware of a bias using internal accuracy assessments, it follows that accuracy cannot be inherent to a dataset.
Looking at the interplay between mathematical notation and language provides more insight. For example, we describe the mathematical symbol with the word mean. We don’t stop there, however; we also sometimes call it the average. Likewise, the mathematical symbol s is described by the words standard deviation, but we also know s as precision, sigma, repeatability, and sometimes spread. English has a wealth of synonyms, giving it an ability to describe that is unparalleled. In fact, it is one of only a few languages that require a thesaurus. This is why it is important to make a clear distinction between the relatively clear world of mathematical notation and the more free-form world of words. Language gives us flexibility and power, but can also confound with its ability to provide subtle differences in meaning.
When we look at the etymology of the word accuracy, we can see that it is aptly named. It comes from the Latin word accuro, which means to take care of, to prepare with care, to trouble about, and to do painstakingly. Accuro is itself derived from the root cura, which means roughly the same thing and is familiar to us today in the form of the word curator. It is fitting language for a process that requires so much care.
When we discuss measurement error we seldom use mathematical symbols; we use language that is every bit as important as the symbols. The word error itself derives from the Latin erro, which means to wander, or to stray, and suitably describes the random tendency of measurements.
Whether we describe it with mathematics or language, error describes a fundamental pattern we see in nature; independent measurements tend to randomly wander around a mean. When the frequency distribution is normal, accuracy from the underlying truth occurs in multiples of √N. Error is the umbrella covering the other terms because it is the natural starting point for any discussion. Because of this, precision and accuracy are naturally subsumed under error, with accuracy further split into internal and external accuracy. By contemplating all of this, we expose the healthy tension between words and mathematical notation. Neither is perfect. Mathematics establishes natural patterns and provides excellent approximation tools, but is not readily available to everyone. Language opens the door to perspective and point of view, and invites questions in a way that mathematical notation does not.
Final Notes
Making sense of GPS error requires that we take a close look at the intricacies of the GPS signal, with particular attention to the ramp up to a normal distribution. It also requires a good hard look at the language of error. Shifting the GPS data back and forth between the frequency-distribution and time domains nicely illustrates the complications imposed by a non-stationary mean. Datasets that are an hour or less in duration do not always increase in accuracy when the measurements are averaged. Averaging may provide a gain, but it is not a certainty. When the non-stationary mean converges to a Gaussian process after an hour or so, we begin to see what De Moivre discovered almost 275 hundred years ago: accuracy increases as the square root of the sample size.
The GPS system is so good that the division of accuracy into its proper internal and external accuracy components is shimmering beneath the surface for most users. It is rare that a set of GPS measurements has a persistent bias, so internal accuracy assessments are usually appropriate. This should not stop us from being careful with how we discuss accuracy, however. Some attempt should be made to distinguish between the two types, and neither should be used interchangeably with precision. What’s more, while accuracy is not something intrinsic to a dataset like precision, it is still much more than just a descriptive word. Accuracy is the hinge between the formal world of mathematics and point of view. Its derivation from N and s in internal assessments stands in stark contrast to the more perspective-driven derivation often found in external assessments. When carrying out internal assessments, we must be aware that we are assuming that the measurements are centered over truth. When carrying out external assessments, we must be mindful of what outside mechanism we are using to provide truth. True to its word origins, accuracy demands careful and thoughtful work.
David Rutledge is the director for infrastructure monitoring at Leica Geosystems in the Americas. He has been involved in the GPS industry since 1995, and has overseen numerous high-accuracy GPS projects around the world.
FURTHER READING
• Highly Readable Texts on Basic Statistics and Probability The Drunkard’s Walk: How Randomness Rules Our Lives by L. Mlodinow, Pantheon Books, New York, 2008.
Noise by B. Kosko,Viking Penguin, New York, 2006.
• Basic Texts on Statistics and Probability Theory A Practical Guide to Data Analysis for Physical Science Students by Louis Lyons, Cambridge University Press, Cambridge, U.K., 1991.
Principles of Statistics by M.G. Bulmer, Dover Publications, Inc., New York, 1979.
• Relevant GPS World Articles
“Stochastic Models for GPS Positioning: An Empirical Approach” by R.F. Leandro and M.C. Santos in GPS World, Vol. 18, No. 2, February 2007, pp. 50–56.
“GNSS Accuracy: Lies, Damn Lies, and Statistics” by F. van Diggelen in GPS World, Vol. 18, No. 1, January 2007, pp. 26–32.
“Dam Stability: Assessing the Performance of a GPS Monitoring System” by D.R. Rutledge, S.Z. Meyerholtz, N.E. Brown, and C.S. Baldwin in GPS World, Vol. 17, No. 10, October 2006, pp. 26–33.
“Standard Positioning Service: Handheld GPS Receiver Accuracy” by C. Tiberius in GPS World, Vol. 14, No. 2, February 2003, pp. 44–51.
“The Stochastics of GPS Observables” by C. Tiberius, N. Jonkman, and F. Kenselaar in GPS World, Vol. 10, No. 2, February 1999, pp. 49–54.
“The GPS Observables” by R.B. Langley in GPS World, Vol. 4, No 4, April 1993, pp. 52–59.
“The Mathematics of GPS” by R.B. Langley in GPS World, Vol. 2, No. 7, July/August 1991, pp. 45–50.
