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Given a map as image, how can I find out which coordinate reference system it has?

The question is general, but here is a specific example.

Below is the map for which I want to find the coordinate reference system:

European inland waterways - 2012

PDF source with much better quality. The map does not specify the coordinate reference system.

How do I find out the coordinate reference system used?

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What you want is the projection, not the coordinate reference system. You need to look at the lines of longitude and latitude and work out from their curvature what possible projections it might be.

For example, the latitude lines are curved, which rules out anything like Mercator or Peters projection. Conic projections [https://en.wikipedia.org/wiki/Lambert_conformal_conic_projection] have straight lines of longitude and curved latitude - I'm having trouble eyeballing the straightness of the longitude lines but load it into a GIS or graphics package and see if the longitude lines are straight. That might indicate a conical projection. Then you need to figure out the parameters of the conic.

If all the lines are curved, you might have something like an azimuthal projection, and then you need to figure out the parameters of that - what's the projection centred on, and so on.

Sometimes its easier to georeference and warp an image to a known set of coordinates. With a good lat-long graticule on the image you know the true lat-long of a lot of points on the image, and you can use a "rubber sheet" deformation to warp the image to any other projection. The more reference points the better.

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    A map projection is just part of a projected coordinate reference system. Ideally you need to identify the ProjCRS including projection, parameters, linear unit, and the geographic CRS used. As a first step--yes, that's identifying the projection. – mkennedy Jul 6 '18 at 21:08
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An image is just an image and it comes with no any specific geographic reference information. So you are not going to "find out" anything in the image itself.

What you need to do is to find the reference systems that are specifically designed for the geographic area of your interest then apply them to the image. An example link is here What is the ideal equal-area projection to map Germany, Switzerland and Austria (DACH)?

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