I have a project in QGIS which involves interpolating several different variables on several different maps, then using the raster calculator to multiply some of the resulting interpolations together. I'm now trying to automate it so I can create this map on a daily basis. What specific libraries must be installed so I can run interpolation automatically (via a python script, or any other method)? Can anyone share an example of how this can be done?

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You could automate some tasks using the QGIS modeler which allows you to build and run models on selected layers like the following:

QGIS modeler

There's a nice tutorial in the following link which goes into quite great detail on how to create a complex model:

Automating Complex Workflows using Processing Modeler.

You could also use the Python Console to call tools from the Processing Toolbox. For example, to see a list of interpolation tools, you could run the following to find all tools with the keyword "interpolat" in their description (some tools contain "interpolation" and "interpolate", hence why "interpolat" is used):

import processing

r.fillnulls - Fills no-data areas in a raster layer using v.surf.rst splines interpolation or v.surf.bspline interpolation--->grass7:r.fillnulls
r.resamp.interp - Resamples raster map to a finer grid using interpolation.--->grass7:r.resamp.interp
r.resamp.rst - Reinterpolates using regularized spline with tension and smoothing.--->grass7:r.resamp.rst
r.surf.idw - Surface interpolation utility for raster layers.--->grass7:r.surf.idw
v.surf.bspline - Bicubic or bilinear spline interpolation with Tykhonov regularization.--->grass7:v.surf.bspline

You can find more information on how to run these commands from the documentation:

Using processing algorithms from the console.

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