I'll second what oikos said, the paper deserved to be rejected by the journal editor without even being sent out to reviewers. The author of the paper doesn't seem to understand the difference between rate and acceleration. He also doesn't really understand the difficulties in trying to calculate the acceleration of an inherently jittery quantity. It is a numerically unstable process that he tried to clean up by smoothing the first order quantity (the rate of increase). but by doing that he also completely ignores the error in the acceleration, which is certainly comparable to (or larger than) the acceleration itself. He then bases all his conclusions around this highly dubious quantity, rather than the more accurate rate of change or just using the temperature itself--which is clearly going UP.
As a scientist I'm surprised that he could make such a fundamental error, which makes me suspicious of his background--he is an (old) oil exploration geologist.
EDIT: It's pretty easy to see by JimZ's answer that he doesn't really understand acceleration either, and neither did the person who wrote the "note" explaining the graph, so let's think about it. First, if we have temperature data over time, and we want to see whether it's warming, that's very easy to do--we just look at later times, and see if the values are greater than at earlier times. It is quite easy to see from the plot that that is true, so that answers the question "..where's the warming?"--it's right there, in front of your face.
Another way to see whether it's warming or not (although less direct) is to look at the first derivative of the quantity, that is, the SLOPE of the temperature curve. If the slope is positive, then it's warming. Now, like many physical quantities, there is a lot of noise in the signal, and the noise makes the calculation of the slope a numerically unstable process. So the author smooths the temperature curve before differentiating. This makes sense, and what do we find when we do that? We find that the green line (the rate of increase) is positive from 1882 until present. That means that it's been warming that whole time. Of course, this is after the low pass filter has been applied to smooth the curve. So once again, that answers the question "...where's the warming?"--again, it's right there, in front of your face.
Note that the editor's note is WRONG, the editor claims that when the green line curves down, that it is COOLING, but that is not right, as long as the green line is positive it is WARMING, not cooling. The author of the paper wants to infer something about whether the rate of warming is going up or down based on the value of the green line (the acceleration), but as I mentioned before, the error in the acceleration is probably comparable to the acceleration itself, so making interpretations from subtle slope changes of the green line is not a valid procedure. However, we can do something along those lines by simply comparing the value of the green curve over a long time period. For example, the value of the green curve now is substantially greater than it was in the mid-20th century, if that difference is significant (the author would need to show that), then we can say that not only is the temperature going up, but the rate of temperature increase is accelerating.