measuring horizon elevation at various bearings

plumbline, protractor and a straight stick

or two measuring sticks and a spirit level

It's 40-odd years since I (very briefly) knew how to use a sextant but won't he need one with an artificial horizon in order to meaure the elevation of the actual horizon above the "natural" horizon? If so, I think that one on eBay may well lack it. And marshy ground doesn't seem a good place for using a reflecting plane - especially not the bowl of mercury I seem to recall using with one ancient sextant ;)

A theodolite measures both bearing and elevation quickly and accurately. Ebay has many, but beware of "antique" Indian made imitations. Something from the 60's to 90's should be both accurate and affordable.

Something like

Use the phones compass to align it and measure the heights on the image later.

Or maybe print it and call it the arts project?

Horizons can be read off some software geographical mapping tools for walke rs, like Anquet's. Lay out a route direction and radius from your origin and it will plot out a graph of elevations along it to 1m accuracy. You d on't even need to go outside in the cold to do it. Not sure if the related O.S. software has this functionality.

rusty

Rather depends on what you mean by "cheap". I don't know whether you can hire theodolites. How many readings are you trying to take? If I wasn't using a theodolite I would certainly start with a good camera tripod as a basis. Do you need to take "scans" from many viewpoints?

An easy way to capture a lot of data would be to take a number of digital photos around 360 degrees and stitch them together with software (much easier using a tripod). Then you could use landmarks (such as peaks) to calibrate the image both for elevation and bearing, using data from maps and google earth. Half a degree accuracy (diameter of moon or sun) sounds just about achievable provided you have enough good "features".