I'm following this article to downscale a satellite image. I have several predictors to assist the downscalign.

The first step is to resample the predictors to make them match the pixel size of the response variable. Before resampling, I applied a Gaussian filter to each of the predictors , and named them x1_02, x1_04, x2_02, x2_04, where x1 and x2 are the names of the predictors, and the number after the underscore indicates the σ (standard deviation) of the Gaussian filter.

The second step is to create linear models between the repsonse and each of the predictor variables, i.e:

lm1 = y ~ x1_02

lm2 = y ~ x1_04 etc.

lm3 = y ~ x2_02 

lm4 = y ~ x2_04 etc.

Thirdly, I have to find the highest R2 for each predictor. For example, the highest R2 for the predictor x1 was when I used the x1_04 predictor. The highest R2 for the predictor x2 was when I used the x2_08 predictor, etc. Then, I build a model using the best predictors, i.e., final_model = y ~ x1_04 + x2_08 + x3_02 etc.

The next step, is to use the final_model parameters to predict the response at the fine spatial scale and to extract the final_model's residuals created at the coarse spatial scale.

Then, I need to downscale the residuals. For downscaling, I use the atakrig package in R. One of the parameters of the discretizeRaster function is the σ (i.e., the standard deviation). By considering the σ, essentially I minimize the point spread function, thus, mitigate the uncertainties of the downscaling process.

My question is, what value should I put on the σ parameter of the function, considering that in the final_model I used predictors that did not have the same σ?



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