How does a sports watch know your altitude?

"How thick the earth is, how high the sky is/How thick the earth is, how high the sky is/The stars are blinking/The moon is drawing a question mark/The comet is dragging a long tail/The rainbow is building a bridge..."

Does anyone still remember the theme song of "Blue Cat's Three Thousand Questions"? Does it reveal my age? When you were a child, did you ever wonder "how high the sky is, how thick the earth is"?

If you usually wear a sports watch when hiking, you may find that during the hike, the watch always seems to know how high you are. How does it do this?

01

Physics vs. geometry

There are two ways to measure height: physical and geometric. Let's talk about the physical approach first:

If you stand in two places and measure different gravity , it means that the two places are at different distances from the center of the earth (center of mass), that is, they are at different heights.

This kind of physical thinking may sound a bit romantic, but for ordinary people, its practical value is almost zero.

Geometric thinking has been widely used from ancient times to the present. Select one or more observation points, measure various angles, lengths and distances, and then construct a suitable triangle to calculate the height. Even with the satellite positioning system (also a geometric thinking), we still use this method to measure the height of the mountain, even Mount Everest.

Mount Everest measurement. Image source: Visual Planet "How do we measure Mount Everest?"

02

With Satellite vs. Without Satellite

As people learned that the earth is a sphere and the scale of measurement became larger and larger, the situation changed. The measured data had to be adjusted according to the curvature of the ground , and the refraction deviation of light caused by different air densities had to be considered . What's more troublesome is that some places can't be measured at all. For example, if you want to measure an island, you stand too far away and the island is submerged below the horizon. If you want to get closer, there is no land.

A space satellite in an eccentric orbit around the Earth. Copyright image, no permission to reprint

Later, the sky became higher, and satellites could help us see the whole picture of the earth, and also allow our sports watches to know how high we are.

Unlike traditional ground measurements, the satellite positioning system requires a spatial rectangular coordinate system , so that any position in space can be expressed by coordinates. In this coordinate system, the origin is the center of the earth, the Z axis is the earth's rotation axis (the movement of the earth's axis is averaged), the X axis points to the meridian, and the Y axis is determined by the right-hand method taught by the math teacher (as shown in the figure below). Therefore, as long as the distance to 4 different satellites is measured, the coordinates of your location can be calculated, and then other useful data can be calculated .

Map: Xu Jingzhong

In this way, the satellite positioning system has indeed solved the original problems of curved surface and long distance, but new problems have arisen: how to know the exact position of the satellite in space? How to accurately measure the distance between you and the satellite?

To be precise, the time used by the ground and the satellite must be consistent . The specific time and the length of each second must be the same, but the time on the ground is based on the rotation of the earth, and when the satellite uses atoms, this must be coordinated. In addition, the satellite is so far away from us and flies so fast, the speed of time passing is different from that on the ground, so corrections must be made according to Einstein's theory of relativity. Then, you have to calculate the satellite's orbit accurately , which involves many physical parameters. Then, the influence of different air densities on electromagnetic waves still exists , which must be considered when measuring distance. In addition, the receiver of the satellite signal itself may be subject to electromagnetic interference, and problems in software calculations can also cause errors .

Copyright image, no permission to reprint

In fact, the accuracy of satellite positioning system measurements is not necessarily higher than that of traditional ground measurements. Therefore, those measurements with high requirements require the combination of the air network (satellite system) and the ground network (ground-based base stations for auxiliary positioning) , and the two types of data are corrected with each other.

03

Satellite positioning and maps

For practical applications, we need to convert the values ​​of spatial coordinates into positions on the map, that is, to make each point have corresponding longitude, latitude, and altitude.

Map: Xu Jingzhong

It is not difficult to determine the longitude and latitude of a point on the sphere (as shown in the figure above), but it is not so easy to determine the height. Because the question is where the height starts .

It is easy for us to think of using sea level as the starting point of altitude, and then extend this benchmark altitude to the entire earth. Yes, China has observed the rise and fall of the Yellow Sea for a long time , and calculated an average sea level as the zero altitude, which is the starting point for altitude measurement.

But the earth is not a perfect sphere, it is a bit flat , and the benchmark sea level should be extended according to the shape that is closest to the earth, and this shape must be expressible by mathematics. After much deliberation, humans chose the ellipsoid .

Removing the Dry Earth from the Ocean

my country's BeiDou system uses the 2000 National Geodetic Coordinate System, in which the default long and short axes of the Earth's ellipsoid are 6378137 meters and 6356752.31414 meters. So, now we have to use this ellipsoid to determine longitude and latitude, which is a little more complicated than the sphere, as shown in the figure below.

Map: Xu Jingzhong

Well, we just need to extend the Yellow Sea elevation to the entire ellipsoid, and then the distance from your location to the ellipsoid is the altitude. Is this really how sports watches calculate it? Not yet, because the satellite positioning system does not use the ellipsoid extended by the Yellow Sea elevation.

04

Uneven sea level

Please think about it, why do we use sea level as the starting point of height? Because the water surface can represent a surface with equal gravity .

Copyright image, no permission to reprint

We know that the gravity distribution of the earth is not uniform, so the global sea level (scientific name: geoid) is high and low under the influence of gravity , and it is not a regular sphere. If you want to know the most accurate altitude, it should be the vertical distance from the point to the sea level. However, the uneven geoid cannot be expressed mathematically, and the satellite positioning system will not use it.

Image source: "Summary of the Basics of Cartography - The Control Basis of Maps"

What does the satellite positioning system use as the starting point for altitude? As mentioned above, the 2000 National Geodetic Coordinate System used by the Beidou system, the ellipsoid (scientific name: reference ellipsoid) set in it is the starting point for global altitude calculation.

Map: Xu Jingzhong

The reason for setting such an ellipsoid is that after years of research and calculation, it is most consistent with the earth's geometry and can represent the global average water level. Obviously, it is not equal to the elevation extension surface of the Yellow Sea in my country.

Comparison between the satellite positioning system measurement and the true altitude. Map: Xu Jingzhong

Now we know that the height we measure through the satellite positioning system, such as the reading on a sports watch, is the ellipsoid height on the left side of the above picture. As for the orthometric height on the right side of the above picture, that is, the pure altitude, you must know where the real geoid is. It coincides with the water surface on the sea surface, but on the land part (especially in the mountains), this surface is invisible and difficult to locate. Therefore, China extends the elevation of the Yellow Sea to the entire land and regards it as height 0, forming an approximate geoid. The heights on the maps and geographical signs you see in daily life are approximate altitudes starting from this .

References:

[1] GPS Principles and Receiver Design, Xie Gang, Electronic Industry Press, 2017.1

Author: Xu Jingzhong and Fan Qi

Review｜Liu Songchuan Senior Engineer, Institute of Aerospace Information Innovation, Chinese Academy of Sciences

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