根据观测到的日出计算二维地形的算法
南美洲的一群人沿着赤道站在彼此等角距离的点上（从地球中心开始测量）。由于山地地形，它们各自的海拔高度不同。我们的目标是使用手表确定他们的海拔。
春分这一天，当太阳从正东升起时，每个人都会聚精会神地等待并记录（以令人印象深刻的精确度和准确度）太阳尖端第一次可见的确切时间（GMT）。对于一些人来说，这是它出现在美丽的南大西洋地平线上的时间。对于其他人来说，这是它从山顶的山脊上窥视的时候。
给定一个将观察者的经度与他们第一次目睹太阳的时间配对的元组列表，您可以对沿赤道的高度的特定采样做出任何具体的声明吗？您是否必须知道第一个观察者的海拔高度（在本例中为海拔 0 英尺，脚趾浸入海滩的水中）？你需要团队的人完全覆盖赤道、环绕式吗？如果你用这几百人的微薄团队都无法解决这个问题，那么你可以用几乎无限数量的观察者来解决它吗？
不，这不是作业问题。
A team of people in South America stand at points along the equator at an equiangular distance from each other (measured from the center of the earth). Due to mountainous terrain, they each stand at different altitudes. Our goal is to determine their elevation using watches.
On the vernal equinox, when the sun rises due east, each person waits attentively and records (with impressive precision and accuracy) the exact time GMT that the tip of the sun was first visible. For some, this is the time that it appeared over a lovely South Atlantic horizon. For others, this was the time that it peeked over the ridge of a mountain top.
Given a list of tuples pairing the longitude of the observer with the time they first witnessed the sun, can you make any concrete claims about a particular sampling of the altitude along the equator? Do you have to know the elevation of the first observer (in this case 0' above sea level, toes in the water on the beach)? Do you need the team of people to completely cover the equator, wrap-around style? If you cannot solve it with this meager team of hundreds, could you do it with a nearly-infinite number of observers?
No, this is not a homework problem.
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对于每一个经度，您都可以计算出当阳光在该位置切向接触地球时的时间，这就是理论上日出的时间。
当太阳升起较早时，观察者显然站在一座小山上。根据他的经度与此时太阳预计升起的点的经度之间的差异，您可以计算出（我认为
（cotangens（phi） - 1）x半径）观察者比现在应该是日出的地方高多少。这给了你观察者高度的下限，因为预计太阳现在升起的点也可能在山上。当观察者看到太阳的时间比预期晚时，他显然是站在山的西侧。从这个时间就可以计算出太阳的高度角。结合东边下一个观察者的计算，你可以得到他的高度上限。
如果每个观察者都能在地平线上（或在东方的山顶上）看到他的一位同事，或者看到地平线上的海洋，你就可以准确地计算出他们的高度。
For every degree of longitude you can calculate the time, when the sunlight touches the earth on this location tangentially, that is the time, when the theoretical sunrise should be.
When the sun rises earlier, the observer obviously stands on a hill. From the difference between his degree of longitude and the degree of longitude of the point where the sun is expected to rise in this moment, you can calculate (I think
(cotangens(phi) - 1) x radius) how much higher the observer is than the place where now should be the sunrise. That gives you just a lower limit of the height of the observer, because the point, where the sun is expected to rise now, could be on a hill as well.When an observer sees the sun later than expected, he obviously stands on the western side of a mountain. From the time you can calculate the altitude of the sun. Together with the calculations of the next observer to the east, you can get an upper limit of his altitude.
If every observer can see one of his colleagues on the horizon (or on top of the hill in the east) or sees the ocean on the horizon, you can calculate their altitudes exactly.
可推导出的事实：精确的 GMT 时间可用于计算在海平面没有遮挡物的情况下太阳与观察者的角度。
为了充分理解这一点，让我们看一下“孤立”的案例。如果一个人向您提供他们第一次看到太阳的经度和 GMT 时间，您只能得出第一次看到太阳的角度。这意味着该角度处有障碍物，但上方没有任何障碍物。我们不知道我们在哪里，有多高，但我们知道相对于我们的位置，远处的某些东西遮挡了太阳
。让我们看一下两个主要情况。第一个人站在海洋中，看到太阳为我们提供了时间和海拔的基线。第二个人站在稍微靠后的地方（可能海拔更高），可能会先看到太阳。
从这里有2个案例。人 2 可以看到人 1：如果是这种情况，则人 1 和人 2 之间的地形没有障碍物。如果第二个人同时或先于第一个人看到太阳，我们就知道这一点。通过三角函数，我们可以确定第二个人的高度变化，因为我们已经创建了一个三角形，并且知道足够的角度和足够的边来解决它。
另一种情况无法解决。如果人2看不到人1，这是因为人2和人1之间有一些地形障碍。我们可以确定人2和人1之间的海拔比他们都高，你可以猜测一下大约地形是什么样子，但你无法确定第二个人的海拔，因为他们可能“就在山脊后面”或“山坡下”
所以几百个可能行不通，但是，如果有足够的观察者，那么每个人人可以看到至少 1 个其他人在他们前面（后面不算），您可以相对准确地确定海拔。
Derivable fact: The exact GMT time can be used to calculate the angle of the sun to the viewer at sea level with no obstructions.
To fully understand this, let us look at the "isolated" case. If a single person gives you their longitude and the GMT time they first see the sun, you can only derive the angle that the sun was first viewed. Which means there is an obstruction at that angle but nothing above it. We don't know where we are, how high we are, but we know relative to our position, something in the distance obscures the sun
Lets look at the two main cases. Person 1 is standing in the ocean, and sees the sun giving us the baseline for Time and Elevation. Person 2 is standing somewhat further back, (probably at a higher elevation) and will probably see the sun first.
From here there are 2 cases. Person 2 can see Person 1: If this is the case, then the terrain between Person 1 and Person 2 is non obstructive. We know this if Person 2 sees the sun at the same time or before Person 1. Via trig, we can determine the elevation change of person 2, because we have created a triangle, and know enough angles and have enough edges to solve it.
The other case is not solvable. If Person 2 cannot see Person 1, this is because there is some terrain obstruction between Person 2 and Person 1. We can determine that the elevation between Person 2 and Person 1 is higher than both of them, and you can make some guesses at approximately what the terrain looks like, but you cannot determine Person 2's elevation because they could be "just behind a ridge" or "down the slope of the mountain"
So a few hundred probably wouldn't work, however, with enough observers so that each Person can see at least 1 other Person in front of them (behind doesn't count) you can determine the elevation with relative accuracy.
太阳以每小时 15 度经度的速度在天空中移动。如果我们将观察者的经度间隔 1 度，第一个观察者位于东部陆地边缘，并且所有观察者都处于海平面且没有障碍物，那么他们将期望以 4 分钟的间隔看到太阳升起，并且他们应该看到太阳在顺序，从东到西。与这些时间的任何偏差都与 i) 海拔或 ii) 海拔和东部障碍物的组合有关。
我们能够确定观察者的绝对海拔：i) 东部没有更高的海拔，或 ii) 如果该位置东部的任何观察者具有更高的海拔（以便这些障碍物），则有足够长的时间返回到海平面或低于海平面不再阻挡通往太阳的道路）。对于这些类型的点，我们可以使用与预期时间的时间差来确定在海平面上看到日出的位置（该位置的东部），并将该切线与垂直于地球的线相交观察者所在位置的表面。该线到与切线相交处的高度将是绝对高程。
对于i）东边更高的观察者或ii）没有足够的无障碍距离来重新建立视线的观察者，如果绝对已知东边点的高程，我们就可以确定真实高程。对于下一个点，只有高于该点我们才能确定绝对高程。很可能在多个点之后，只能确定相邻点的相对高程，而不是它们的绝对高程。这是因为太阳升起的时间受到海拔和东方障碍物高度的影响。
其他重要的考虑因素/问题：
因此，我认为答案是，尽管在理论上可以知道每个点的真实海拔，但在给定自然地形的情况下，不一定可以确定每个点的绝对海拔。我认为观察者的数量并不重要，这只是采样间隔和分辨率的问题。我认为在整个赤道周围都有观察员也没有帮助。希望这能引起一些反思:)
The sun moves across the sky at a rate of 15 degrees of longitude per hour. If we space our observers 1 degree of longitude apart, the first being at the eastern land edge, and all were at sea level without obstructions, then they would expect to see the sun rise at 4 minute intervals, and they should see it rise in order, from east to west. Any deviations from these times are related to either i) altitude or ii) a combination of altitude and obstructions to the east.
We are able to determine the absolute elevation for observers that have i) no higher elevations to the east or ii) a sufficiently long return to sea level or below if any observers to the east of the location had a higher elevation (so that those obstructions no longer block the path to the sun). For these types of points, we can use the time difference from the expected time to determine the location where the sunrise would have been seen at sea level (to the east of the location), and intersect that tangent with a line perpendicular to the earth's surface at the observer's location. The height of that line up to where it meets the tangent will be the absolute elevation.
For observers that have i) higher observers to the east or ii) insufficient obstruction-free distance to re-establish the line of sight, we can determine the true elevation if the elevation of the point directly to the east is known absolutely. For the next point, we can determine the absolute elevation only if it is higher than this point. It is likely that after a number of points, it will only be possible to determine the relative elevation of neighbouring points, not their absolute elevation. This is because the time at which the sun rises is affected by both elevation and the height of the obstruction to the east.
Other important considerations/questions:
Therefore, I think the answer is that it is not necessarily possible to determine absolute elevation at every point given natural terrain, although there are theoretical terrains in which the true elevation of each point could be known. I don't think the number of observers matters, its only a question of sampling interval and resolution. I don't think that having observers around the entire equator helps either. Hope this can cause some reflection :)