In QGIS, I'm trying to georeference an image where lines of latitude and longitude are curved / non-parallel (see below) and I have no information about the projection used. Can anyone recommend the transformation settings I should use and the CRS?

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About the CRS use EPSG:3034, ETRS89-extended / LCC Europe. It is recommended by the Information and Service System for European Coordinate Reference Systems (http://www.crs-geo.eu/crs-pan-european.htm) and I think it is probably the CRS used in the original map.

About the transformation type use Thin Plate Spline. The image seems to be pretty deformed, so may be difficult to adapt with a "polynomial" transformation (TPS can be represented also as a polynomial transformation, I am referring to the other polynomial ones). Use as many control points as you can, including the limits of the image extent. Think in the Thin Plate Spline algorithm as a triangulation between control points, defining a linear transformation for each triangle. Control Points will be mapped exactly to their pair.

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