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I have two layers: One a modeled dataset with points on a 1x1km grid across the globe. I also have another layer which is my own data points which are scattered across the globe. What I want to do is find the nearest modeled value from the 1x1 grid data and append that to my table for my data points.

I thought it would be a join function but as the points are not from the exact same coordinates I need some way of extracting the nearest point to my sample point.

Can someone help me with which tool to use and how best to use it?

I am using ArcGIS and Windows 10.

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You can use spatial join in the toolbox, or join based on spatial location (see the first drop down menu in the "join" form (right click > join and relate > join).

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