I'm a global warming skeptic, and one of my concerns is the accuracy of historical global temperatures. Since these temperatures weren't sampled at random locations or gridded points, can they be considered an accurate representation of "global temperature"?
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3982162/ talks about this in a temporal sense, since using the average of the minimum and maximum daily temperatures at a given location isn't really a great way of determining average temperature. However, it doesn't discuss geographical random sampling/gridding.
http://onlinelibrary.wiley.com/doi/10.1002/joc.4580/full notes this problem exists, and takes uniformly gridded measurements, but it's limited to a specific region and time period (1979-2012).
I know climate scientists slice the Earth up into grids to avoid clustering bias, but that's not the same thing, and isn't useful if the original readings don't accurately represent the slice/region.
I also realize that climate scientists have other measures of global warming, but linear regression of actual temperature measurements seems to be the most used to convince the public, so their accuracy seems important.
As a skeptic, I'd also like to know if, in general, most of the arguments for global warming are statistical in nature (ie, linear regression on measured variables), or the statistical ones are just the most "photogenic" for public consumption? In other words, is the whole non-randomly-sampled/gridded temperature argument a red herring?
EDIT (to clarify question):
To determine the Earth's mean surface temperature, we can employ one of these methods:
Measure the Earth's temperature at every point and average. Of course, this is physically impossible, since a point is a 0-dimensional mathematical abstraction, but we can do something close with satellites.
Select a large number of random points on the Earth's surface (this random distribution is uniform in longitude, but not in latitude-- in latitude, it would look like a cosine curve), measure the temperature, and average. In addition to giving us a mean, it would give us a standard deviation so we can say "we are 95% confident that the Earth's true mean temperature is X plus or minus Y".
Take a uniformly spaced grid (non trivial, since the distance between longitudes vary by latitude), measure the temperature at those points and average. This is similar to the first approach, but with fewer points. Unless we believe our grid points introduce a bias, this should be as accurate as random sampling.
My problem: temperature measurements in the past were made using NONE of these methods. The points where temperature was measured were not chosen randomly or in a gridded fashion. Therefore, how can they be an accurate measurement of historical temperature, even if we only consider temperature changes?
NOTE: I realize surface temperature isn't the best measure of global warming, since water has a much higher specific heat than land (among other things), but that's my focus for this question.