## Critical Points

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0.

The above is from Wikipedia. As you all should know, I know Calculus. Now, technically, no climate data is a truly differentiable  function, since they are not provided in continuous form. Nevertheless, one can estimate the instantaneous rate of change of a function from slopes of lines between discrete points. So with timeseries that are not continuous in a mathematical sense, one can nevertheless identify points at which the rate of change abruptly switches sign. I am looking into this as a way to identify intervals of change for better estimating variance adjustments for LT versus surface temperatures. So my interest is in finding something analogous to “critical points” in the surface or LT data, specifically ones associated with changes in the sign of temperature change. So I will want to identify when, say, three month slopes change sign, and then identify those intervals between them that are useful for assessing short term amplification. I am just getting started, but will start with a plot of identified intervals in the surface data (here I will be using an average of GISS, HadCRUT, and NCDC over 1979-almost the present (deviations from the 1981-2010 mean), and three month averages to make the points of change stand out more. Note that later when finding the magnitude of temperature changes I will probably just use the unsmoothed data over identified intervals). The plot shows those intervals I chose to be sufficiently long, taking breaks in the intervals that were short as not seperate intervals put part of the longer intervals, and making them overlap on the maxima/minima that characterize the change in rate.

Now, having identified intervals, I examine the ratio of changes in the troposphere over said intervals: two problems occur: Near zero the ratios vary a little too much, and also a couple of times the rates of change are opposite in sign. This can be seen in a plot of the ratios with the the amount of atmospheric change as the independent variable:

See, now, a deceptive thing I could have done was not tell you that my new approach was failing to overcome the noise. But I am showing you this plot so you can see why I am going to say: I am disappointed, because this is not the kind of clean result I was hoping for, showing the short term amplification without a lot of noise. So anyway, a couple of things that might help: Eliminating ratios associated with changes too close to zero, and unphysical ratios less than zero. The result (eliminating ratios where the absolute value of the atmospheric change less than .1)  is that the mean ratio goes from about .64 to 1.04 and the standard deviation goes from .81 to .46, so while it does converge closer to be within the realm of my previous estimates (which were roughly from 1 to 1.5 with a best estimate of about 1.3, very close to the theoretical/model value of 1.2) it clearly is very uncertain. So I would consider this not a great method to get the right value for the amplification ratio for short term changes. I am still most satisfied with taking various different approaches and averaging them. So remember our previous estimates: 1.46, 1.07, 1.49, 1.32, 1.49, .954, 1.56, and 1.38, and add in the new estimate, the average is ~1.3 so a conservative estimate of the short term amplification remains about 1.2, with some uncertainty still to be figured out. (Excluding the estimates based on the smooth trend removal method, since it is a little difficult to explain and I am not sure it is a detrending method that is reliable anyway, the new mean is 1.28)

So I feel I have not made much progress in the variance adjustment estimate. Since I don’t yet have a good method to get it, I am going to be faced with the issue of continuing to search for a good method. Well, for now, I have the best I can do.