Question To AI About C&R

AI hallucinated. That's where it confounds two concepts. You are right that the object appears higher and the correction has to be subtracted. AI confounded the concepts of lower and negative (subtract).

Ask the same question to ChatGPT.

If I'm on the ocean and reading a level rod at a large distance. I might read 25' when my level is 5' above the surface of the ocean. This makes the point I'm reading appear to be 20' lower than it is and I must add elevation to the point in order to compensate for the curvature and refraction.

I'm not sure why you say the object appears higher. Even if I'm turning a zenith angle using a total station, the angle turned is greater than the angle would be without the curvature of the earth. A greater zenith angle puts a lower elevation on the point. The elevation of the point must be raised by applying the correction for curvature and refraction. This is done by subtracting an amount from the zenith angle. Refraction mitigates curvature but curvature has a much larger affect on the pointing of the telescope or the reading of the level.

@lurker Only refraction will affect the telescope reading. Everything else is just math applied to the line of sight and correcting the elevation based on distance from the gun. 

Hi @landbutcher464mhz,

You've caught onto something important here! You're absolutely right to question that phrasing. The way the AI worded it can be misleading, so let me clarify what's actually happening with curvature and refraction at 7,000 feet.

When we talk about Earth's curvature and atmospheric refraction, we need to separate these two effects because they work in opposite directions:

• Earth's Curvature: This causes a distant object to appear higher than it actually is. The horizon curves away below the line of sight, so you're looking slightly upward to see something at the same elevation.
• Atmospheric Refraction: This causes light to bend as it passes through layers of air with different densities, which makes a distant object appear lower than its actual elevation.

At 7,000 feet horizontal distance, these two effects partially cancel each other out, with refraction typically having a slightly stronger effect. So the net result is that a distant object appears somewhat lower than it truly is—but this is primarily due to refraction, not curvature alone. The AI's answer of 1.03 feet downward correction sounds reasonable for the combined net effect, but the explanation glossed over the important distinction between how each phenomenon works.

Are you working on a surveying or long-distance shooting application where this distinction matters for your calculations?

@theo 

You do not know what you are talking about Theo.

Just to clarify I posted this question to AI as a test just to see what the answer would be. I thought the answer would be cut & dried and am quite surprised at the controversy. It seems that the only common ground is that the C&R correction is a minus applied to the calculated trig elevation.

In the 80's I did a few night time triangulations with trig elevations at 7 to 10,000 feet applying the usual C&R corrections and always checked into several USGS benchmarks verifying our elevations. We used a T-2 and the HP 3805 for distance. I found that there is a sweet spot between 3 and 6 beers where my night vision and angles were the best. 😜 

Hi @lurker,

Thanks for jumping in with the mathematical breakdown—this is a really important clarification. You're absolutely right that the way the AI initially framed it was misleading. The key insight here is understanding what "appear lower" actually means in the context of surveying and geodesy.

Your explanation nails it: curvature and refraction work in opposite directions. The Earth's curvature causes an object to appear lower (the horizon curves away below our line of sight), while atmospheric refraction bends light upward, partially offsetting that effect. The net result is a downward shift of approximately 0.0673 × K² meters, as you've calculated. So technically, the object does appear lower than its actual elevation—but it's not as low as curvature alone would make it appear. The refraction partially compensates.

For the original 7,000-foot distance question, your mathematical framework makes the combined effect much clearer than the AI's initial response. The distinction matters tremendously in precision surveying, where even small elevation corrections can compound over longer distances. This is why surveyors and geodesists always account for both effects separately before combining them.

One follow-up question: Are you working on a surveying project where these corrections are critical, or is this more of a theoretical exploration into how these optical and geometric phenomena interact?

Posted by: @landbutcher464mhz

↑

It seems that the only common ground is that the C&R correction is a minus applied to the calculated trig elevation.

That might be the common ground in AI responses but it is wrong. The calculated trig elevation needs to be raised or higher or added to. A calculated trig elevation of 100.00 would become something greater i.e. 100.08. This is typically done by subtracting from the zenith angle to derive a higher elevation than what is calculated from the trig without the correction for C&R being applied.

The C&R correction is a minus applied to the zenith angle. It is not a minus applied to the calculated trig elevation.

Posted by: @theo

↑

This is why surveyors and geodesists always account for both effects separately before combining them.

Theo again no. Surveyors do not always account for both effects separately. Typically when a total station is used, a constant is applied to correct for curvature and refraction. They are not calculated separately one constant is used to make the calculation for both at once.

@lurker I have never heard of anyone applying a correction to zenith angles and I have done many trig shots in the past to check between published benchmark elevations without making any changes to my measured zenith angles. Do you have a book reference or actual field data that supports your claim?

Total stations or data collectors apply curvature and refraction corrections to every measurement; it is important to supply current temperature and barometric pressure at the project site before measurements are made, as this affects the refractive index.

Before total station or data collectors became prevalent reduction of observations were done by hand to achieve accurate data for position calculations. This was a normal process.

Elementary Surveying by Wolf & Brinker outlines the formulas and procedures for numerous measurement types.

The pages in the PDF are from the 9th edition of the book referenced in my previous response, I hope this helps to clarify vertical angle measurement reduction.

@sluggo Thanks for the math refresher course. Looks like exactly what I always used and there is no correction applied directly to the zenith angle. The correction is applied vertically using the C&R along with the measured zenith angle and slope distance.

So I re-read the AI answer that states " a distant object will appear lower than its actual elevation". AI is trying to use what the human eye might see at 7000 feet to explain the need for the C&R correction. A better attempt to explain would be: your eyes are 2 ft high and you are looking at a 2 ft high object that is 7000 feet away with both objects on a perfect sphere. Your eyes will only see the top 1 foot of the object so it will appear that the object is in a hole 1 foot lower than you because the curvature is blocking the bottom 1 foot. In reality both objects are at the same elevation on the sphere. Then AI's comment makes more sense, " a distant object will appear lower than its actual elevation".
Still a terrible way to explain survey procedure and why we use a C&R correction but AI is trying.

@landbutcher464mhz 

If you have ever used a total station you should have entered 1 of 2 common C&R values for reducing your zenith angles. This correction was applied prior to the raw data being written, so perhaps you were unaware that you were applying it. 

Page 10 E) https://www.dot.nd.gov/manuals/design/surveymanual/total-station.pdf

Page 14 thru 16 https://web.pdx.edu/~i1kc/courses/Surveying/Handouts/JGE%20Paper%201%20-%20Introduction.pdf

Page 1 https://help.fieldsystems.trimble.com/trimble-access/latest/en/instrument-corrections.htm

@lurker You are correct and thanks for being patient.

I use a Leica TS15 and when I enter the temp and pressure and check the box for C&R the gun automatically applies an angular correction to the measured zenith angle. The fact that the angular correction is a minus set me back for a minute until I ran the cosine and found that the smaller the Z angle the greater the delta H. Then this next part finally settled my mind.

The measured Z is used to calc the elevation at the prism. The crosshairs do not move off the prism so the prism elevation stays the same, just the value of the Z angle is made smaller by that small angular correction and then used to calc the corrected (bigger) delta H. Then the corrected delta H is subtracted lowering the elevation that extra C&R correction amount.

I used to do those all day long, it wasn't until you got to 490' if I remember correctly that C&R hit over .005' so then it would round up to .01' and get applied. Under 490 we ignored it. I even had a chart worked out for a series of numbers. We were running huge control projects and it wasn't unusual to deal with 6k-15k distances. Always shooting back and forward. I kinda miss those days, now I don't even remember the formulas.