This page is where I will post follow-ups and revisions to my recent article The Great Climate Shift of 1878, posted on WattsUpWithThat?
Reader Greg Goodman had a number of constructive suggestions, one of which was to remove the oscillatory modes from the data and run a fit to equation 2 on the residual. This removes any reliance on the SSA for extraction of the trend and thus precludes the possibility that the observed response is SSA related. If the fit parameters come out close to that which was achieved by fitting equation 1 to the differenced trend, it will be confirmation of the result.
Here is the hadcrut4 NH data with the 2 main oscillatory modes removed:In order to use equation 2 on the data, we need to add a constant of integration, C and solve for the boundary condition of continuity at the step. The modified equation 2 is:
Eqn 2a – Modified 2nd order system response allowing for non-zero initial condition and with boundary condition of continuity at t=τ
Fitting Eqn 2a to the data of figure 1 yields the following parameters:
Compared to the previous fit obtained by fitting eq1 to the differentiated SSA mode.
We see the new fit yields a slightly more damped system with a slightly higher natural frequency. Given the data of figure 1 is much noisier, especially near t=tau, and the difference in detection method, I believe these results are confirmative. Of significance is the match in τ, the time at which the shift occurs.
Figure 2 shows the two responses
The fit is shown below (left) along with the residual (right)
Adding the oscillatory SSA modes to eqn (2a) is shown below against the unaltered dataThe correlation coefficient R= .8476 is insignificantly better than the R=.8471 achieved with the original model using the integrated eqn 1. The correlation is dominated by the unpredictable events in the data. Smoothing the data with a HP smoothing factor of only 5 yields R= .971 for both models.
The plot below shows a three harmonic sine fit to the two SSA main modes. Note how the combined SSA modes (red) seem to change during the step response transition region (t=27-50). The three sine waves comprising the fit are shown on the right (fundamental period = 209 years). All three components are near their peaks. Time to get out your winter woollies for the next few decades.
Update: Monday,Oct. 7
I’ve done the above analysis on the SST data.
The eqn 2a parameters are
Update: Monday,Oct. 7 PM
If we take equation 2a above to represent a system whose output is the temperature anomaly, then that output is the convolution of the input forcing with eqn 2a. The plot below is the cyclical convolution of the yearly sunspot number with eqn 2a, scaled by a gain constant (.0005) and offset by .9 degrees.
The correlation of the convolution with the data is excellent until about 1970 until the kernel end-of-record effects become apparent.
This result is strong empirical evidence both for the skill in which equation 2a encapsulates the climatic response and for the sun spot number being a primary forcing function.
[Note: In order to improve readability I have removed some earlier updates which were in error or incomplete ]
Update Tuesday, Oct 8, 8PM
Here’s the result of matching the kernel length to the SSN data compared to Gregs data (red) and the hadcrut4 SST data (gold).
In the plot below, the system response delay taken out.
Update Tuesday, Oct 8, 3:45AM
The plots above were convolved against a kernel derived from the SST NH data. In the plot below, the system response was derived as above (parameter fit of eqn 2a) but using Greg Goodman’s adjusted SST data. The correlation with the data is excellent until about 1963. The apparent cooling (shaded area below) might be explainable by volcanic activity [although Greg’s not buying it – see his comments below].
The graph below is Figure 1 in the last IPCC report. The black line shows temperatures. The four biggest tropical eruptions over the past century had slight cooling effects.
Update Tuesday, Oct 8, 8PM
The plot below adds in a single mode (SSA[L=81,k=3,4]) to the convolved SSN recordI had to reduce the kernel damping factor from the derived .135 to .5 to get the last 25 years to match. Here’s why the damping factor came in so low: As the plot above shows, the trend plus just a single SSA mode describes Greg’s adjusted SSA data (R=.963). When we remove the mode from the data we get: Fitting eqn 2a to what is essentially a sinusoid results in a low damping factor.
One difference is that I have been using unsmoothed monthly data decimated to yearly by sampling each June. The rationale for this strategy can be seen by looking at the SST stack plot.
Each column is a month. June has the least variance which I suppose is due to calmer seas. Since we want to avoid filtering to preserve as much transient behavior as possible and since the winter variance is uninteresting to the model, we sample in June. Greg’s stack plot is more homogeneous which indicates that perhaps the data was smoothed across months.
In either case (low damping + volcanic cooling / higher damping and no effect from volcanoes), it points out how much the Hadcrut adjustments have biased the IPCC conclusions regarding AGW. If Greg’s “unadjustment” is correct, one glance at the SSA analysis disproves any notion of AGW. And one glance at SSN convolution shows what really controls the climate. The sun, whoda thunk?
Update Wednesday, Oct 9, 6AM
The plot below shows the residual error of the model (SSN convolved with eqn 2a + mode 2).
Update Wednesday, Oct 9, 8AM
Here’s a backcast of the model. The assumption is that mode 1 above (the 58 year cycle) remained constant which may not be true. I don’t know how to check this since it is a SST record but it seems to match the BEST land temp record pretty well.
Update: Friday, Oct. 11, 3PM
Appended Greg’s SSN prediction to data set. Convolved with unmodified kernel.