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split lines with polygons

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Hallo!

I have a set of vessel path lines and am trying to calculate marine traffic density on a grid, or preferably on a polygon fishnet. At first I tried using the line density tool, but that does not work, since it calculates density in such a way, that I can't get the exact number of lines crossing each mesh cell.

Currently I am trying to split the lines by a polygon fishnet and then count the number of lines within each polygon using the spatial join tool. I used the Intersect tool to split the lines, but that does not give me what I want, since it splits the lines also at intersections between the lines themselves (the path lines often cross each other), not only at the intersection with polygon boundaries.

Help would be much appreciated.
Gašper
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