How does GPS work?

GPS satellites are equipped with an extremely accurate atomic clock and know their positions at all times. They continually transmit the time and their location using radio waves. A mobile phone or other navigation device receives these signals from all satellites within their line of sight. The difference between the arrival time at the local clock of the receiver and the transmission time recorded by the satellite clock corresponds to the time taken for the signal to travel from the satellite to the receiver (the “time of flight”). Since radio waves travel at the speed of light, the time of flight determines the distance covered by the signal. The satellite positions and the distances are used to calculate the position of the receiver using a system of equations.

This simplified description does not take into account the fact that the local clock in the receiver is not an atomic clock. If it is inaccurate by just one millionth of a second, the calculated position will be inaccurate by at least 300 meters. The GPS problem is the need for the phone or other navigation device to determine the precise time along with the location — known in the theory of relativity as space-time.

If too few satellites are in the line of sight, the system no longer functions reliably and delivers multiple solutions, in other words several different locations where the receiver could be. This can lead to the situation where a phone indicates an incorrect location or no location at all. Until now the number of satellites needed to obtain unique solutions to the GPS problem has only been conjectured.

Five satellites for a precise location

Mireille Boutin, a professor of discrete algebra and geometry at TU/e and Gregor Kemper, a professor of algorithmic algebra at TUM, have now produced a mathematical proof showing that with five or more satellites, the exact position of the receiver can be uniquely determined in almost all cases. “Although this was a long-standing conjecture, nobody had managed to find a proof. And it was far from simple: We worked on the problem for over a year before we got there,” says Gregor Kemper. At present every location on Earth has sight contact to at least four satellites at all times. “Roughly speaking, with only four satellites, the probability of having a unique solution to the GPS problem appears to be 50 percent. Proving that statement is one of our next projects,” says Kemper. With three or fewer satellites in the line of sight, GPS navigation definitely does not work.

Geometry and uniqueness

The researchers arrived at the proof by characterizing the GPS problem in geometric terms. They found out that the position of the receiver cannot be uniquely determined if the satellites are located on a hyperboloid of revolution of two sheets. This is a curved surface that is open in all directions. Although this result is theoretical, it has the practical benefit of offering a better understanding of inaccuracies in determining positions.

Dr. Zihao Ou, assistant professor of physics at The University of Texas at Dallas, is lead author of the study, published in the Sept. 6 print issue of the journal Science.

Living skin is a scattering medium. Like fog, it scatters light, which is why it cannot be seen through.

“We combined the yellow dye, which is a molecule that absorbs most light, especially blue and ultraviolet light, with skin, which is a scattering medium. Individually, these two things block most light from getting through them. But when we put them together, we were able to achieve transparency of the mouse skin,” said Ou, who, with colleagues, conducted the study while he was a postdoctoral researcher at Stanford University before joining the UT Dallas faculty in the School of Natural Sciences and Mathematics in August.

“For those who understand the fundamental physics behind this, it makes sense; but if you aren’t familiar with it, it looks like a magic trick,” Ou said.

The “magic” happens because dissolving the light-absorbing molecules in water changes the solution’s refractive index — a measure of the way a substance bends light — in a way that matches the refractive index of tissue components like lipids. In essence, the dye molecules reduce the degree to which light scatters in the skin tissue, like dissipating a fog bank.

In their experiments with mice, the researchers rubbed the water and dye solution onto the skin of the animals’ skulls and abdomens. Once the dye had completely diffused into the skin, the skin became transparent. The process is reversible by washing off any remaining dye. The dye that has diffused into the skin is metabolized and excreted through urine.

“It takes a few minutes for the transparency to appear,” Ou said. “It’s similar to the way a facial cream or mask works: The time needed depends on how fast the molecules diffuse into the skin.”

Through the transparent skin of the skull, researchers directly observed blood vessels on the surface of the brain. In the abdomen, they observed internal organs and peristalsis, the muscle contractions that move contents through the digestive tract.

The transparent areas take on an orangish color, Ou said. The dye used in the solution is commonly known as FD&C Yellow #5 and is frequently used in orange- or yellow-colored snack chips, candy coating and other foods. The Food and Drug Administration certifies nine color additives — tartrazine is one — for use in foods.

“It’s important that the dye is biocompatible — it’s safe for living organisms,” Ou said. “In addition, it’s very inexpensive and efficient; we don’t need very much of it to work.”

The researchers have not yet tested the process on humans, whose skin is about 10 times thicker than a mouse’s. At this time it is not clear what dosage of the dye or delivery method would be necessary to penetrate the entire thickness, Ou said.

“In human medicine, we currently have ultrasound to look deeper inside the living body,” Ou said. “Many medical diagnosis platforms are very expensive and inaccessible to a broad audience, but platforms based on our tech should not be.”

Ou said one of the first applications of the technique will likely be to improve existing research methods in optical imaging.

“Our research group is mostly academics, so one of the first things we thought of when we saw the results of our experiments was how this might improve biomedical research,” he said. “Optical equipment, like the microscope, is not directly used to study live humans or animals because light can’t go through living tissue. But now that we can make tissue transparent, it will allow us to look at more detailed dynamics. It will completely revolutionize existing optical research in biology.”

In his new Dynamic Bio-imaging Lab at UTD, Ou will continue the research he started with Dr. Guosong Hong, assistant professor of materials science and engineering at Stanford and a corresponding author of the study. Ou said the next steps in the research will include understanding what dosage of the dye molecule might work best in human tissue. In addition, the researchers are experimenting with other molecules, including engineered materials, that could perform more efficiently than tartrazine.

Study authors from Stanford, including co-corresponding author Dr. Mark Brongersma, the Stephen Harris Professor in the Department of Materials Science and Engineering, were funded by grants from federal agencies including the National Institutes of Health, the National Science Foundation and the Air Force Office of Scientific Research. As an interdisciplinary postdoctoral scholar, Ou was supported by the Wu Tsai Neuroscience Institute at Stanford. The researchers have applied for a patent on the technology.

“Bats have gained a bad reputation as being something to fear, especially after reports of a possible linkage with the origins Covid-19,” says study author Eyal Frank, an assistant professor at the Harris School of Public Policy. “But bats do add value to society in their role as natural pesticides, and this study shows that their decline can be harmful to humans.”

Frank compared the effect of bat die-offs on pesticide use in counties that experienced those bat population declines to counties that were likely unaffected by the wildlife disease. He found that when the bat populations declined, farmers increased their use of pesticides by about 31 percent. Because pesticides have been linked to negative health impacts, Frank next tested to see if the increased use of pesticides corresponded with an increase in infant mortality — a common marker to study the health impacts of environmental pollution. Indeed, when farmers increased their use of pesticides, the infant mortality rate rose by almost 8 percent. This corresponds to an additional 1,334 infant deaths. Or, for every 1 percent increase in pesticides, there was a 0.25 percent increase in the infant mortality rate.

The study also found that pesticides aren’t as good at preventing pests as bats. The quality of the crops likely declined, as farmers’ revenue from crop sales decreased by nearly 29 percent. Combining this revenue loss with the expense of the pesticides, farmers in communities that experienced the bat die-offs lost $26.9 billion dollars between 2006 and 2017. Adding onto those losses the $12.4 billion in damages from infant mortality, the total societal cost from the bat die-offs in these communities amounted to $39.6 billion.

“When bats are no longer there to do their job in controlling insects, the costs to society are very large — but the cost of conserving bat populations is likely smaller,” says Frank. “More broadly, this study shows that wildlife adds value to society, and we need to better understand that value in order to inform policies to protect them.”

A team of data scientists at eBird, the participatory science platform, has packaged summaries covering every bird species, in every state, and made them available online for free. These data summaries will help states prepare their federally required 2025 updates to State Wildlife Action Plans.

“As we began to work more closely with state agencies and regional conservation partnerships, we realized that we needed to significantly increase the accessibility of eBird information for these partners,” said Viviana Ruiz-Gutierrez, assistant director of the Cornell Lab’s Center for Avian Population Studies and the driving force behind development of the state summaries.

“By providing these customized summaries, state agencies don’t have to wrangle with big data and spatial tools. They get data targeted to the area they are responsible for,” said Andrew Stillman, a postdoctoral fellow at the Cornell Lab. “It’s much more efficient, saving them time and money.”

State Wildlife Action Plans are critical to conservation in the United States, Stillman said. The plans must be updated every 10 years and submitted to the U.S. Fish and Wildlife Service for approval. Approval releases funding from the State and Tribal Wildlife Grants program, which is used to proactively conserve birds and other species that make up the biodiversity of each state.

The 2025 updates will mark the second major revision to state wildlife plans since the first plans were completed in 2005. But this is the first time eBird state data summaries will be available to inform the revisions, helping planners easily identify which species are in greatest need of conservation and to set priorities for where and when to take conservation action.

Without year-round weekly bird abundance data from eBird, an important part of the big picture is missing. For example, tundra swans don’t breed in Michigan and are not found there for most of the year. But during two weeks in March, 13% of the global population is migrating through Michigan, making marsh and wetland habitat vital for stopovers during their long journey back to their Arctic breeding grounds.

The state summaries are updated each year with new population numbers from eBird. With the latest August 2024 update, planners can now also see which way bird populations are trending for the entire state: increasing, decreasing or stable; and by how much.

“We’ll continue to refine and update the summaries so states have what they need,” Stillman said. “We’re also looking into expanding this customization for the two dozen Migratory Bird Joint Ventures in the U.S. and Canada. Birds are not known for recognizing human boundaries and joint venture partnerships work across boundaries to conserve birds and the habitats they need, where they need it. The state planners tell us, ‘Keep it coming.'”