Time for a recap, hopefully this will help clarify the development thus far. First, a diagram of the model we’re developing.

Figure 1: SST is the convolution of the climate impulse response, h(t) with the SSN, where SSN is the sun spot number - 49.534 plus a 58+ year decadal oscillation (SSA mode 1)

Figure 1: SST is the convolution of the climate impulse response, h(t) with the SSN, where SSN is the sun spot number – 49.534 plus a 58+ year decadal oscillation (SSA mode 1)

The circle with the “X” inside represents the convoltion operator. h(t) is the impulse response of the climate system, which is given by the equation. The a and b parameters were set to give the correct step when we fit the step response here:
stepSST

The ω,ζ parameters were found by fitting a second-order response characteristic to the step response above. This first cut estimate is close enough to determine the response lag used in the parameter optimization procedure below. Here is the raw convolution of the SSN data (with the fake data appended to make the correlation easier to see) with the unoptimized h(t).

Raw correlation output vs. adjusted SST data

Figure 3 – Raw correlation output vs. adjusted SST data. System response parameters estimated from step response.

I’ve left off any reference to dates in the plot above because we haven’t yet determined the delay of h(t). The discretized kernel is full length (in this case, equal to the the number of SSN data points including the fake ones) but it will have an unknown effective length because the samples at the beginning of the filter contribute negligibly to the output sample (think of the filter as a weighted sum with the weighting coefficients increasing as we move toward the output sample). We can by eyeball, see the obvious correlation but the data has to be “back-ed up” by 18 years (the effective filter delay) to match.

Raw convolution delay matched to data. Effective delay of h(t) = 18 years

Figure 4 – Raw convolution delay matched to data. Effective delay of h(t) = 18 years

Let’s define the singular spectrum analysis modes we will be using to complete the model.

SSA mode definitions

Figure 5 – SSA mode definitions

The non-linear lowest order “mode” is traditionally referred to as the trend in the literature. The trendless oscillatory signals are reconstructed from the eigenmodes (k values) grouped by frequency. The reconstruction built from the non-periodic modes (or periodic signals whose amplitudes are trivial) will be referred to a the residual. Note that since in SSA, trend + modes + residual = data, the residual may sometimes be refer to data – trend – mode, as the two usages are mathematically equivalent.

From this point on, the results presented will be slightly different (and slightly better) than previously shown. Due to some confusion on my part about the start and end dates of the adjusted SST data, I had used a 20 year delay instead of the correct 18 derived above. I have also changed the way I find the optimal values for the system response parameters ω and ζ. Instead of fitting the convolution results to the noise data minus the primary SSA modes as before, I have instead fit to the the SSA trend plus the modes 1-3. This avoids biasing from the spiky events in the climate record that are presumed to be unrelated to the h(t) response.

I’ve also changed the fit algorithm to minimize the variance rather than maximize the correlation because the correlation is insensitive to the arbitrary scaling constant (i.e the kernel “gain” that converts sun-spot numbers to temperature). The results with the optimized system response parameters are shown below shown below.

SSN data convolved with h(t) + modes 2,3 and 4 versus adjusted ICOADS SST data

Figure 6 – SSN data convolved with h(t) + modes 2,3 and 4 versus adjusted ICOADS SST data

The plot below shows the results of the convolutional part of the model with no additional oscillatory modes added.

Figure 7 – SSN convolved with h(t), no oscillatory modes added
Upper left: convolution vs data.
Upper right: oscillatory modes (none)
Lower left: residual
Lower right: PSD of residual

With only the primary 58.4 year mode1 added the results improve markedly.

SSN convolved with h(t), SSA{L=81,k=3,4] mode added Upper left: convolution vs data. Upper right: oscillatory modes (none) Lower left: residual  Lower right: PSD of residual

Figure 8 – SSN convolved with h(t), SSA{L=81,k=3,4] mode added
Upper left: convolution vs data.
Upper right: oscillatory modes (none)
Lower left: residual
Lower right: PSD of residual

Mode 1 is the is the so called “60 year cycle” (here 58.4 year) often identified with the Atlantic Multidecadal Oscillation (AMO). It’s existence is controversial but it needs to be stressed at this point that the conclusion presented here is independent of the MDO signal. Whether it exists as a periodic forcing or is a statistical fluke, figure 7 shows that the convolution of the mean-centered sun-spot data (which is obviously free of any anthropogenic affects) with h(t) accounts for 90% of the observed temperature anomaly and the remainder, plotted as the residual in figure 7, is trendless and free of any detectable AGW signal. Thus the entire SST temperature record is free of any anthropogenic contribution.

Alarmists will no doubt raise the curve-fit canard. Any such critic demonstrates a lack of understand of system dynamics. The system impulse response, h(t), defines a linear time-invariant (LTI) filter. It’s output at any point in time, is a weighted linear combination of the input samples accumulated up to that time, with the weighting factors fixed and predetermined by the response parameters defined above. Such systems cannot create information, they can only process the information present in the input signal. The fact that any such LTI system function exists which when convolved with the SSN input signal produces the observed temperature trend proves that all of the information required to explain that portion of the observed record is contained in the input signal.

Physical Implications
The system response function h(t) contains two integrations. As such, its output is sensitive to asymmetry in its input measured over roughly a 36 year (twice the effective filter lag) time period. That is, when the mean-centered SSN data (presumably a proxy for TSI) double averaged over approximately 36 years is positive, the temperature trends up. If negative, down. In fact, h(t) can be approximated as an accumulation (pure integration) of an 18 year moving average, scaled and offset as shown in the figure below.

.25+ΣMA(SSN,18)/1800

Figure 9 – .25+ΣMA(SSN,18)/1800 vs SST-SSA[L=80,k={3,4}]


Figure 9 alone should but to rest any notion that this is some sort of curve fitting exercise.

h(t) describes a system that stores energy during warm periods and gives it back over cool periods. I am not a climatologists but one can imagine an ocean transport mechanism that could exhibit this behavior.

Direct TSI Influence
Because climatologists have ignored the climate’s integrative properties, they have discounted the TSI effect as being too small to account for the observed temperature anomaly. Indeed, the system response h(t) function greatly attenuates the 11 year solar cycle due to the double integration noted above. However, Mode 3 of figure 5 shows there is an 11 year cycle present in that SST data. It is at the .03-.05K amplitude others have found and is nearly in-phase with the SSN data. This phase relationship shows that it is a direct heating effect, bypassing the system response shown above. The figure below adds mode 3 to the model, improving the correlation slightly.
figure9

There is also some correlation (r=.24) between the residual above and the averaged annual ENSO index.

Residual from figure 9 vs scaled ENSO index, Raw (left), Hodrick-Prescott (L=500) filtered (right)

Figure 12 – Figure 11 -Residual from figure 10 vs scaled ENSO index, Raw (left), Hodrick-Prescott (L=500) filtered (right)

Extending the model to the present
Because of the delay inherent in h(t) we can’t determine the present output. Greg Goodman had the great idea of extending the SSN data using the prediction of the solar cycle available from ??. Since most of the data in the kernel pipeline is real and since the first samples are lightly weighted, we expect this to work pretty well for 18 years or so. Figure 12 shows this to be the case.

Figure 13 - Model extended to present using projected SSN data

Model extended to present using projected SSN data


Figure 13 - Modeled versus Adjusted SST; SSN data extended with projection

Figure 12 – Modeled versus Adjusted SST; SSN data extended with projection

Keep in mind the blue curve of figure 13 (the h(t) output) is independent of the red data. Note how the peaks in the ripple of the model align with the el-nino events present in the data record.

I’m publishing this incomplete post because I’ve had trouble with wordpress crashing and losing my drafts. Watch this page for updates

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