Create stage height raster using least cost path and r

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I have a point shapefile of Station IDs and stageheights. I would like to create a raster where each cell has the stage height value (in meters) of the closest in situ station to that cell.

I want this raster to match up with another raster. So I would like it if I could input both a raster I have created (dataset 3 described below) and my point shapefile (1).

Datasets:

1) Point Shapefile with stage heights of a river delta

2) Shapefile of the river delta extent

3) Raster of the delta where NA's represent land (could also have them be zero's if need be) and 1's are water. Two datasets 10 meter resolution and 30 meter resolution.

One conceptual issue I am having is with the amount of small streams I have. For example (pictured in image below), station 1 (circled in blue) is technically closer to the black x region than station 2 (circled in red), but the stage height value in red is more representative of point x. There are NA's in between the two streams, does that mean that the value will not jump across streams?

Conceptual Issue Image

How can I reassign the values in my Raster (all the 1's) to the stage height of the nearest station and make sure that these values are not jumping from stream to stream? Do I need to use least cost path? What is the best way to do this?

I would like to use R, but can use ArcMap if I must.

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SeldomSeenSlim On

So I'm not sure what tools you have available to you but I think this answer may be useful:

Calculating attribute for network distance between multiple points in ArcGIS Desktop?

Here the questioner was looking to calculate distances on roads to some points, but your problem seems similar. I think the main point I would make here is that you should do your network distance classification prior to worrying about the raster layer. You may have to convert from polygon to lines or some workaround to get your data into a format that works, but this is the kind of job the tool is designed to do.

After you have reclassified your river shapefile based on their network distance to a given point, then convert the polygons to raster and use this to classify your original raster. You could do this in R or Arcmap. Arcmap will probably be faster.