What exactly is “acceleration”?

Hey, I’m still alive people. Well, today’s topic is pretty dull, “technical stuff” but I can’t think of much else to do at the moment. I am over do for examining the satellite trends, I guess I’ll do that later.

We have talked before about the idea that there have been “accelerations” in  various climate parameters that are also ongoing at least according to some. But how does one define “acceleration” exactly? Well first let’s deal with the question of instantaneous acceleration. A time series at any given time may be increasing or decreasing. By analogy to motion of a projectile mass, this motion has “velocity” represented by it’s first derivative. If the time series is decreasing, the “velocity” has a negative value, if it is increasing the “velocity” has a positive value. Acceleration is occurring at a time when the time series is decreasing or increasingly faster with time, or the second derivative is positive for increasing values or negative for decreasing values. Many people miss define “acceleration” by simply say it is when the second derivative is positive, which is completely wrong. Acceleration is when the first and second derivatives have the same sign. This however is not generally the problem with claims of “acceleration” in climate data. That is a problem of average acceleration. You see, time series are not continuous functions, and true differentiation requires continuous data. But more than that, statistical “fitting” of data allows one to determine “trends” in the data and derivatives there of, but when done over an entire data set can tell you if there was an overall increase in the rate of change (which is, again, not the same thing as acceleration) not whether that change is ongoing. For instance, the following illustrative diagram shows an “accelerating” time series:

The graph isn’t of anything in particular, just some numbers. The rate of change is clearly not constant. Someone could conceivably “differentiate” this data and then fit a line to the derivative and conclude that, since the average rate of change is positive and the slope of the fitted line is positive, the data is “accelerating” the only problem being that they would be flat wrong. True, acceleration did take place in this data, but not continuously and ongoing, and the data is not “currently”, that is, at the end of the series, “accelerating”.  Previously we have shown that datasets are not currently undergoing acceleration by pointing out that similar length periods with the same rates of change have occurred in the earlier part of the data set, but really all that would actually be necessary is to show that rate may have accelerated at some points but is not presently doing so. In the case of GMST it most certainly isn’t, as the analysis involving comparisons we have done more than adequately demonstrates. But what about claims of acceleration in Ocean Heat Content, for example? Well, looking back at the data from that post (undoubtedly an updated analysis would look different) you can see where claims of acceleration can get misleading. The rates of change in that data set went from negative early on to positive, and not withstanding some ups and downs, stabilized to fairly constant plateaus from which it dropped and rose back to a couple times. The most recent period had the rate completely flat, which suggests that the data were not showing current and ongoing acceleration. This distinction can hardly be emphasized enough. Hopefully this post will help people more carefully look at this question in the future.