Adding one more
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If the exposure itself is very smooth, like distance from a factory, then it is generally not possible to identify effect of such spatially smooth exposures on spatial outcomes due to the possibility of unmeasured smooth spatial confounding, and identification would need additional assumptions.

If your spatial exposure varies at fine spatial scales, then confounding due to unmeasured variables that are such smooth(er) functions of space can be mitigated by use of standard spatial estimators like those based on Gaussian process, generalized least squares, splines etc.

I think you are referring to what is known in spatial statistics as 'spatial confounding', that is spatial exposure and spatial outcomes are partly collinear with smooth functions of space (in your case, distance to the center). See some recent work by our team on this.

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GLS estimates can also adjust for spatial endogeneity without explicitly modeling the correlation between the exposure and spatial error. This gives an example where an estimator based off an exogenous model can account for endogeneity. 4/4

Generalized least squares (GLS) estimates with Gaussian process working covariance are consistent for the linear exposure effect under unmeasured spatial confounding, as long as the exposure has some non-spatial component. This overturns claims that GLS fails under confounding. 3/

Restricting spatial random effects to be orthogonal to the exposure doesn’t resolve spatial confounding bias. Restricted spatial regression yields the same effect estimate as unadjusted OLS, with asymptotically non-vanishing omitted variable bias. 2/