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:

$\begin{array}{cc} \{ & \begin{array}{cc} \frac{a e^{\zeta \omega (\tau -t)} \left(\left(2 \zeta ^2-1\right) \sinh \left(\sqrt{\zeta ^2-1} \omega (t-\tau )\right)+2 \sqrt{\zeta ^2-1} \zeta \cosh \left(\sqrt{\zeta ^2-1} \omega (t-\tau )\right)\right)}{\sqrt{\zeta ^2-1} \omega }+a \left(-\frac{2 \zeta }{\omega }+t-\tau \right)-b t+\mathbb{C} & t\geq \tau \\ \mathbb{C}-b t & \text{True} \\ \end{array} \\ \end{array}$
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:
$\omega \to 0.0655,\zeta \to 0.772,a\to 0.0137,b\to 0.00851,\mathbb{C}\to 0.0711,\tau \to 26.8\$
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

Figure 2 – Comparison between eq1 fit to differenced SSA trend vs. eq2a fit to data of figure 1

The fit is shown below (left) along with the residual (right)

Figure 3 – Fit over data (left)
Residual (right) with (blue) and without (red) Hodrick-Prescott M=5 filter

Adding the oscillatory SSA modes to eqn (2a) is shown below against the unaltered data

Best-fit eqn (2a) + SSA[L=82,k=3-6] vs. Hadcrut4 NH data

The 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.

SST – SSA[L=82,k={1,1}]

The eqn 2a parameters are $\omega \to 0.0684,\zeta \to 0.359,a\to 0.00855,b\to 0.00461,\mathbb{C}\to -0.0933,\tau \to 29.87$

The fit to the differenced trend is shown below.

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.

Convolution of system kernel formed from eqn 2a with yearly sunspot number (red) vs. SST NH data

The correlation of the convolution with the data is excellent until about 1970 until the kernel end-of-record effects become apparent.

Residue from above

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 record

Convolved SSN + SSA[L=81,k=3,4]  R=.941

I 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:

SSA[L=81,k=1,2],SSA[L=81,k=3,4] (right)

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.

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

Here are the remaining modes not included in the model.

Beyond mode 5, the reconstructions are noise-like.

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.

Backcast of model including mode 1

Update: Friday, Oct. 11, 3PM

Appended Greg’s SSN prediction to data set. Convolved with unmodified kernel.

Convolution with forecasted SSN appended.