  Adjusting for an unmeasured confounder is generally an intractable problem,
but in the spatial setting it may be possible under certain conditions. In this
paper, we derive necessary conditions on the coherence between the treatment
variable of interest and the unmeasured confounder that ensure the causal
effect of the treatment is estimable. We specify our model and assumptions in
the spectral domain to allow for different degrees of confounding at different
spatial resolutions. The key assumption that ensures identifiability is that
confounding present at global scales dissipates at local scales. We show that
this assumption in the spectral domain is equivalent to adjusting for
global-scale confounding in the spatial domain by adding a spatially smoothed
version of the treatment variable to the mean of the response variable. Within
this general framework, we propose a sequence of confounder adjustment methods
that range from parametric adjustments based on the Matern coherence function
to more robust semi-parametric methods that use smoothing splines. These ideas
are applied to areal and geostatistical data for both simulated and real
datasets
