Many Flat Earther have problems to understand what an apparent horizon is and how it is created. I'll try to explain it with the following animation. The animation below was rendered using my Refraction Simulator.

The left scene shows no refraction. The targets are in a distance of 2, 3, 4, 5, 7.5 and 10 km from the observer and the observer height is 1.5 m.

The middle animation shows different looming conditions from a refraction coefficient of k = 0 to k = 1.9. Standrd refraction is about k = 0.17. If refraction is k = 1 then light gets bent along the surface and the earth looks flat. The horizon raises to eye level and we cannot assign a distance to the horizon, because is is virtually at infinity. If k > 1 light gets bent more than the radius of the earth. The horizon appears above eye level far in the distance and the earth appears concave.

The right graph shows the temperature profile (temperature gradient) and the corresponding refraction coefficient as a function of altitude. We can see that even very small temperature changes can cause remarkable refraction.

What is a Water Horizon?

The water horizon appears as a horizontal line between water and the background. It appears where the surface of the water changes from sloping up to sloping down away from the observers point of view. In a 2D orthographic cross-section side view this is the point where a line from the observer touches the water surface of the curved earth tangentially. We can calculate the distance between the horizon line and the observer using the equations below.

Is the Horizon something physical? Yes. For a water horizon to appear you need water and water is physical. There is a physical location where your personal horizon appears. Someone can go there and mark your horizon line. Boats can sit on your horizon. But the horizon is also a personal thing. Each person has its onwn horizon, because each person has its own altitude and location.

Like a rainbow, you can never reach your horizon, because its locations depend on your position and altitude. The horizon recedes away from you as you try to reach it. But someone else can physically go there, touch it and place marks along your horizon line. But this line is not a horizon for this person.

So the horizon is a virtual line on the water, whose position depends on your position and altitude and refraction, as we will see later. Although the horizon line is a virtual line, that you can't touch, it is also a physical location, where objects can sit. You can touch the horizon line of another observer, but not your own horizon. For the person who touches your horizon, it is not a special place. It's a place like any other place on the water for him.

The horizon is a physical location in the distance, that is personal to every observer and depends on its location, altitude and refraction.

Geometric Horizon

The cyan line in the animation above markes the vertical location of the geometric horizon with respect to eye level. This is the horizon that can be calculated from geometry assuming straight light rays, i.e. without taking refraction into account. The geometric horizon has no meaning for an observer if refraction is not zero. We could see the geometric horizon with a device that receives a sort of light that is not affected by refraction.

The distance to the geometric horizon can be calculated as follows:

In our image without refraction the horizon is according to (1) at 4.372 km. So the third target (the second black target) is just before the geometric horizon distance.

Refracted Horizon

As light gets refracted by the atmosphere, mostly down, distant objects appear loomed up with respect to their physical position. This is also true for any position on the water. Looming appears as an increase in the size of the planet. As the surface of the water gets loomed up increasingly with increasing distance, the apparent horizon recedes away from the observer. The new loomed horizon is not the same physical location as the not loomed geometric horizon. The new loomed horizon is the appearance of a line that lies behind the geometric horizon of the earth.

We can calculate the new distance to the apparent loomed horizon with the same equations above by increasing the radius R depending on the refraction coefficient k:

Note: this equations only work für k < 1. If k ≥ 1 then the horizon is at infinity or the earth looks concave and the horizon cannot be calculated easily.

If you send a friend out to mark the locations of your loomed horizon, he would mark locations that are farther away than the location of your geometric horizon. So if you compare the marks of your horizon from a day with no refraction to a day with looming conditions, you will see that the loomed marks are physically behind the not loomed marks.

You can see this effect in the animation above. The more looming is applied, the more hidden ground comes into view from behind and builds a new horizon. The old horizon locations are no horizon locations anymore.

Again: the refracted horizon is a physical location someone can go and mark. It is not an absolut location that anyone can go and mark. It is your personal location. You have to tell the other person, where to go to pyhsically mark your personal horizon. Then you can go there and say: this markers mark the locations where I saw my horizon from a certain place at a certain refraction.

You can measure the locations of this horizon markers e.g. with GPS. It will give you the geometrical locations of the markers. But from the distance, due to refraction, you saw this markers at another apparent position, at your personal horizon. So the apparent positions of objects in the distance are displaced images of objects sitting at geometric locations on the physical earth. This physical locations are fixed and do not depend on refraction or the observer. But it depends on refraction and your location whether they appear at your horizon or not.

Can we measure the geometric Positions of Objects?

If refraction apparently displaces objects in the distance, how can we measure where their real position is on earth?

We have to use non-optical methods, or apply optical methods in a way refraction cancels, or we have to measure refraction so we can correct for refraction. All this methods are in use. Today mainly the first one using Differential GPS.

If we established the real geometric positions of objects, we can calculate refraction from the difference of the apparent position and the geometric positions.

In The Rainy Lake Experiment we measured the geoometric locations of a set of targets in different distances with Differential GPS. Then we took images through a theodolite and auto level to measure the apparent positions of the targets. From the difference we could calculate the refraction.

Because GPS gives the geometric 3D positions in space in cartesian coordinates with the origin at the mass center of the earth, the locations of the target water levels give the real geometry of the frozen lake they are planted on. If the lake were flat, all water level vectors would lie on a flat plane. But the GPS measurements clearly show that the lake is curved. We could even measure the radius of the earth from this curvature and it matches what is published for this location.