I have tried downloading FRAGSTATS but this crashes and does not work for some reason.
Is it possible to measure some simple fragmentation metrics on a raster landcover map in ArcGIS Desktop instead?
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The FRAGSTATS software is functionally dead and will likely stay that way. The developer Kevin McGarigal has retired and is not looking back. There is nobody that has taken up the reins in maintaining the software.
There use to be an ArcGIS add-in called PatchStat but it operates on polygons and I believe is now defunct. There is a GRASS GIS module, r.li that is written in C++ and has numerous metrics available. The one caveat is that it is limited to raster output using a focal metric approach. This means that landscape-level and class-level metrics are not available.
The current drop-in replacement for FRAGSTATS is the landscapemetrics R package. There is also the R package landmetrics but, it is only available on GitHub. Both of these packages allow for evaluating a given metric around a point location as well as providing focal metrics. The
landscapemetrics package provides the original FRAGSTATS hierarchy of landscape-level (entire raster), class-level and patch-level metrics.
There is also a Python library pylandstats that could be plausibly ported for use in ArcGIS.
You do need to be specific on what metric you are after and should also be aware that there is no one metric for fragmentation, it is inferred by multiple metrics.