If you usually have the habit of climbing mountains with a sports watch, you may also find that on the way to climbing, the watch always seems to know how high it is now?

01

Physical thinking vs geometric thinking

There are two ways of measuring height: physical and geometric. Let's start with physical thinking:

If you stand in two places and the measured gravity is different, that is, the distance between the two places and the center of mass is different, that is, the height is different.

This physical idea sounds a bit romantic, but for ordinary people, the practical value is about zero.

The idea of geometry has been widely used since ancient times. Select one or more observation points, measure the data of various angles and long distances, and then construct a suitable triangle to calculate the height. Even if we have satellite positioning systems (also geometric ideas), we will still use this method to measure the height of mountains, even Mount Everest.

02

Satellite vs no satellite

As people know that the earth is a ball and the scale of measurement is getting larger and larger, the situation has changed. The measured data should be adjusted according to the curvature of the ground. At the same time, the deviation of light refraction caused by different air density should also be considered. What is more troublesome is that some places cannot be measured at all. For example, if you want to measure an island, stand too far away, the island is submerged below the horizon, and you want to get closer, but there is no land.

Later, the human sky became higher, and satellites could help us see the whole picture of the earth, and also let our sports watches know how high the position is.

Unlike traditional ground surveys, satellite positioning systems require a spatial Cartesian coordinate system so that any position in space can be represented by coordinates. In this coordinate system, the origin is the center of the earth, the Z axis is the rotation axis of the earth (the motion of the earth axis takes the mean value), the X axis points to the meridian, and the Y axis is determined according to the right-hand method taught by the math teacher (as shown in the following figure). So, as long as you measure the distance to four different satellites, you can calculate the coordinates of your location, and then calculate other useful data.

In this way, the satellite positioning system does solve the original surface and long-distance problems, but new problems arise, how to know the exact position of the satellite in space? How to accurately measure the distance between you and the satellite?

In order to be accurate, the time used by the ground and the satellite must be the same first. Now, the specific time and how long each second should be must be the same. However, our ground time is based on the rotation of the earth, and when the satellite uses atoms, it must be coordinated. Also, the satellite is so far away from us and flies so fast that the speed of the passage of time is different from that of the surface, so it should be corrected according to Einstein's theory of relativity. Then, you have to be precise about the orbit of the satellite, and there are a lot of physical parameters involved here. Then, the influence of different air density on electromagnetic waves still exists, which should be considered when ranging. Also, the satellite signal receiver itself may be subject to electromagnetic interference, software calculation problems will also cause errors.

In fact, the accuracy of satellite positioning system measurement is not necessarily higher than the traditional ground measurement. Therefore, those demanding measurements need to be combined with the air network (satellite system) and the ground network (base station for ground-assisted positioning), and the two data are mutually corrected.

For practical application, we need to convert the value of spatial coordinates into the position on the map, that is, let each point have the corresponding longitude, latitude and height.

It is not difficult to determine the latitude and longitude of the points on the sphere (as shown in the figure above), but determining the height is not so simple. Because where the height starts is a problem.

It's easy to think of using sea level as a starting point for height, and then extend that reference height to the entire planet. Yes, our country has been observing the fluctuation of the Yellow Sea for a long time and calculated an average sea level as zero altitude, which is the starting point for altimetry.

But the earth is not a positive sphere, it is a little flat, the reference sea level should be extended according to the shape closest to the earth, and this shape must be expressed mathematically. After much thought, humans chose ellipsoids.

The geoid, whose enclosing body can be approximated by an ellipsoid

China's Beidou system uses the 2000 national geodetic coordinate system, in which the default long half axis of the earth ellipsoid is 6378137 meters and the short half axis is 6356752.31414 meters. So, now to determine the latitude and longitude, we should also use this ellipsoid, which is a little more complicated than the normal sphere, as shown in the following figure.

Well, let's just extend the elevation of the Yellow Sea to the entire ellipsoid, and then the distance from your position perpendicular to this ellipsoid is the height. Is that really how it works in the sports watch? Not yet, because the satellite positioning system does not use this ellipsoid extended by the Yellow Sea elevation.

04

uneven sea level

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

We know that the earth's gravity is not evenly distributed, so the global sea level (scientific name: geoid) is high and low under the influence of gravity, not a regular sphere. If you want to know the most correct altitude, it should be the distance from the vertical to the sea level. However, the uneven geoid cannot be expressed mathematically, and satellite positioning systems will not be used.

What does the satellite positioning system use as the altitude starting point? As mentioned above, the 2000 national geodetic coordinate system used by the Beidou system, and the specific ellipsoid (scientific name: reference ellipsoid) is the starting point for the global calculation of altitude.

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

Satellite positioning system measurements and pure altitude distinction contrast. Drawing: Xu Jingzhong

Now we know that the height we measure through the satellite positioning system, such as the reading on the sports watch, is the height of the ellipsoid on the left side of the picture above. As for the positive height on the right side of the above figure, that is, the pure altitude, you must know where the real geoid is. The sea surface part coincides with the water surface, but the land part (especially in mountainous areas), this surface is neither visible nor difficult to locate. Therefore, our country extends the Yellow Sea elevation to the entire land as a height of 0 to form an approximate geoid, the height on the maps and geographical indications you see everyday is the approximate altitude starting from this.