The notch in the Sun-Earth relationship is the dog that didn’t bark — the clue that was there all along, telling us something about the way the Sun influences Earth’s climate. There is a flicker of extra energy coming in at the peak of every solar cycle — roughly every 11 years. It’s only a small peak, but there is no warming on Earth at all — it’s like the energy that vanished. A good skeptic would be saying *but, the increase in energy is so small, how could we find it among the noise*? And the answer is that Fourier maths is so good at doing this that it is used every day to find the GPS signals which (as David details below) are so much smaller than the noise that they are much harder to find than this signal from the Sun.

Thousands of engineers know about and use Fourier maths and notch filters, but due to a strange one-sided bureaucratic funding model, none of those thousands of experts have applied that knowledge, which is so well adapted to feedback systems to the Sun Earth energy flows. David has used an input-output “black box” method to find the empirical transfer function and discover the notch. Viva the independent scientist, supported only by independent donations — at a fraction of the cost of the billion dollar models, David Evans has done something in three years which none of the bureaucrat-driven golden icons have managed in thirty years.

Is the notch real? It shows up independently in different eras and different datasets (see fig 2). What does it mean? Something is occurring which is “tuned” to the solar cycles to change the way the Earth reacts to incoming solar radiation just as that radiation peaks. The mechanism for this must originate on the Sun, because the timing is too accurate, and that unknown mechanism obviously has an influence on Earth’s temperature (maybe through clouds). Obviously to build a full climate model we need to understand that.

Notch filters are used in electronics to filter out “hum”. Notch filters usually do not involve a delay, but they could, which alerted us to the possibility of a delay. This eventually led to the discovery of an 11-year (or half solar cycle) delay, originating on the Sun, between the solar peaks in sunlight and the factor that neutralizes their effect. David discusses a few possible mechanisms in a later post. He finds evidence suggesting that that this indirect effect of the Sun is ~14 times more powerful at driving changes in our climate than the influence of variations in direct solar heating. Something about the Sun, some force, is changing conditions on Earth in a way that conventional climate models don’t understand. They are “plugging the gap” with CO2 in the last 50 years, but can’t possibly work until they understand this missing key. In this post we start the hunt.

Thank you to those who keep us going. Together we hope to advance our understanding of what controls the climate…

— Jo

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# 21. The Notch in the Empirical Transfer Function

Dr David Evans, 27 November 2015, Project home, Intro, Previous, Next.

This post begins the search for the cause of global warming. This is the most mathematical post of the solar part of this series.

We start by finding the empirical transfer function from total solar irradiance (TSI) to surface temperature — which tells us how much surface warming we get per increase in TSI, at each frequency. We find an unexpected notch therein, and discuss its implications.

### Our Formal System

For simplicity while searching for a relationship between TSI S and the surface temperature *T*_{S}, we assume that the TSI is the *only* influence on *T*_{S}. As discussed in the previous post, it is plausible that TSI is a dominant cause of global warming — or more precisely, contains information about a dominant cause because the mechanism is indirect. If TSI mostly predicts *T*_{S}, and there is a strong and obvious relationship between them, then this assumption is adequate for the exploratory analysis here.

Formally, consider the system whose input is the TSI anomaly at 1 AU (the change in TSI at the average distance of the Earth from the Sun, one astronomical unit), denoted by Δ*S*, and whose output is the mean global surface temperature anomaly, Δ*T*_{S}. Both Δ*S* and Δ*T*_{S} are functions of time.

Figure 1: The formal system under consideration: surface warming 100% controlled by changes in TSI.

### Some Background on Transfer Functions

We need to explain the empirical transfer function of interest here, and what it means. This can get technical, but you don’t really need to know the details to understand the implications for climate.

#### – The Transfer Function of Interest

The empirical transfer function discussed below is the ratio of surface warming to change in TSI, or how much surface warming we get per unit of increase in TSI, *by frequency*.

Electromagnetic radiation behaves independently at each frequency in a linear environment such as air or space, which is why we talk of UV, visible light, microwaves, infrared and radio waves and know they do not interact or interfere with each other. Keep on decreasing the frequency, from millions of cycle per second for radio waves down to fractions of a cycle per year, and we get the fluctuations in radiation coming off the Sun on climate time-scales. The formal system above is linear, so the behavior at each frequency is independent of other frequencies, even for the fluctuations on climate time-scales.

Our system is assumed to be linear, because we are only dealing with small perturbations to a stable climate near steady state, and any classical physical system is approximately linear for sufficiently small perturbations. The climate is widely assumed to be linear for warming influences during an interglacial. The system is also assumed to be (time-) invariant, that is, its properties have not changed significantly during the current interglacial.

#### – Transfer Functions Generally

Linear invariant systems have a special property. If the input is a **sinusoid** (see the systems document) then the output is also a sinusoid at the same frequency and, for a given system at a given frequency, the ratio of the amplitude of the output sinusoid to the amplitude of the input sinusoid is constant, and the difference between the phases of the output and input sinusoids is also a constant. Linear invariant systems are amenable to Fourier analysis: the input can be expressed as a sum of sinusoids using the Fourier transform, and then, by the linearity of the system, each of the input sinusoid maps to the same output sinusoid that it would if the input sinusoid were the only input present — that is, what happens at each frequency is independent of what happens at other frequencies. This is the only significance of sinusoids in analyzing systems.

A linear invariant system is completely described by its **transfer function**, which is a function of frequency whose value at each frequency consists of the amplitude multiplier and the phase shift caused by the system. Complex numbers are used to represent sinusoids: both complex numbers and sinusoids have amplitudes and phases, and a sinusoid is simply represented by the complex number with the same amplitude and phase. At a given frequency, the value of the transfer function is the complex number whose:

- Amplitude is the amplitude of the output sinusoid at the frequency divided by the amplitude of the input sinusoid at the frequency.
- Phase is the phase of the output sinusoid at the frequency less the phase of the input sinusoid at that frequency.

The value of the transfer function at a given frequency is the ratio of the complex number representing the output sinusoid to the complex number representing the input sinusoid, using complex division. The Fourier transform of the system output is the complex product of the transfer function and the Fourier transform of the system input.

### The Data

We used the most prominent public datasets, on all time spans available. The datasets are noisy and sometimes contradictory, so we did not pick a “best” dataset or combine them into a composite dataset, but instead found a single spectrum that best collectively fitted all their individual spectra .

The TSI datasets used are Lean’s reconstruction from sunspots from 1610 to 2008 with the Wang, Lean, and Sheeley background correction, the PMOD satellite observations from late 1978, the Steinhilber reconstructions from Be10 in ice cores going back 9,300 years, Delaygue and Bard’s reconstruction from C14 and Be10 from 695 AD, the f10.7 solar radio flux from 1947, and the SIDC/SILSO (V1) sunspot counts from 1749. ACRIM satellite data from 1978 was omitted because it disagrees with the other data before 1992, which then leaves only two sunspot cycles — too short to establish its spectrum.

The temperature datasets used are the satellite records from late 1978 (UAH and RSS), the surface thermometer records from 1850 or 1880 (HadCrut4, GISTEMP, and NCDC/NOAA), the two comprehensive proxy time series of Christiansen and Ljungqvist 2012 going back to 1500 with 91 proxies and 1 AD with 32 proxies, the Dome C ice cores going back 9,300 years (to match the period for which there is TSI data), and Moberg’s 18-proxy series from 1 AD.

### Low-Noise Fourier Analysis

We are interested in two functions of time: the TSI anomaly Δ*S*, and the surface warming Δ*T*_{S}. If those functions were known perfectly for all time then we could calculate their Fourier transforms and learn the amplitude and phase of all of their constituent sinusoids. Instead our data samples these functions, imperfectly, to form a number of overlapping time series of limited extents, from which we estimate the main constituent sinusoids in the functions.

The potential signals we are looking for are small, at about the noise level in the data. Consider the direct heating effect of the TSI peaks that occur at the maximum of each sunspot cycle. From the diagram of the sum-of-warmings model (Fig. 1 of post 13), the direct surface warming due to a change in TSI of Δ*S* is

where *α* is the albedo (~0.30), *λ*_{SB} is the Stefan-Boltzmann sensitivity (0.267 °C W^{−1} m^{2}), *M* is the ARTS multiplier (2.0 [1.5,2.7]), and *f _{α}* is the albedo feedback to surface warming (0.4±0.5 W m

^{−2}°C

^{−1}).

TSI typically varies from the trough to the peak of a sunspot cycle by ~0.8 W m^{−2} out of 1361 W m^{−2}, thus causing ~0.1 °C of surface warming, about the same as the 0.1 °C typical error margin in modern temperature records. However processing techniques like Fourier analysis that correlate an expected signal against the data can easily see beneath the noise floor — for example, the global positioning system signal is typically one four hundredth of the noise floor of the Earth (−26 dB, power).

There are many ways of applying Fourier analysis to estimate the constituent sinusoids. We took care to use methods that minimized the introduction of noise. In particular, we did not arbitrarily change the data in the time series by adding data points whose values are zero in order to “pad” the data series to a convenient length, or use “windowing” — both often done automatically by software packages that apply a Fast Fourier transform (FFT). We originally discovered the notch (below) using the standard Discrete Fourier transform (DFT), by matching a TSI time series to a temperature time series covering the same period with the same number of regularly-spaced data points. But while it is possible to detect the notch using the DFT, we wanted to be surer; so we developed a superior low-noise version of the DFT, called the Optimal Fourier transform (OFT). The OFT has far greater sensitivity and frequency resolution than the DFT, mainly because it is not confined to the preset frequencies of the DFT — but it takes much longer to compute. All the results here use the OFT.

To be properly confident in a Fourier analysis, it must not vary significantly if minor changes are made to the processing methods or to the data — for instance, removing the initial or final 5 or 10% of a data series. The results here are all robust with respect to minor changes in processing technique or data.

### The Empirical Transfer Function

The various datasets each contributed points to a combined amplitude spectrum of either TSI or temperature. The combined spectra were each smoothed, then their ratio found. The result is the amplitude of the empirical transfer function, the black line in Fig. 2. Repeating the procedure, but with the data restricted to pre-1910, to pre- or post-1945, to pre-1970, or to instrumental data (no proxies), gives the colored lines in Fig. 2. The TSI data is at 1 AU so it is deseasonalized, whereas the eccentric orbit of the Earth means the actual TSI incident on the Earth is seasonalized, so we ignore frequencies greater than one cycle per year.

Figure 2: The amplitude of the empirical transfer function, when the data is restricted as marked. The black line is for all data (unrestricted); the gray area is a zone around the black line. The second horizontal scale is the period of the sinusoids. Both scales are logarithmic.

The data was only sufficient to estimate the amplitudes of the constituent sinusoids in the TSI or temperature (or equivalently, their power spectra — the power of a sinusoid is the square of its amplitude). Estimations of phases were not robust. Although the amplitude spectra of physical phenomena like radiation tend to be relatively smooth functions of frequency, the phase spectra are often highly discontinuous so smoothing and averaging is not appropriate.

### The Notch

The first robust feature of the empirical transfer function is the notch, a relatively narrow region of lower response centered on a period of ~11 years. Sinusoids in Δ*S* with periods around 11 years are severely attenuated when transferred by the system to Δ*T*_{S}, relative to sinusoids at other frequencies. Peaks in TSI and sunspots, which occur ~11 years apart on average, do *not* result in corresponding peaks in the surface temperature.

In electronic audio equipment, a filter that removes the hum due to mains power is called a notch filter, because it removes the sinusoids in a narrow range of frequencies around the mains frequency. It appears that something is removing the 11-year “solar hum” from Δ*T*_{S}.

The notch is a curious fact. Solar radiation warms the Earth, providing all the heat as incoming radiation — visible light, UV, infrared, and so on. So we’d expect the peaks in TSI from the Sun every ~11 years to produce small but detectable corresponding peaks in surface temperature; yet they don’t. (This observation gave rise to the notch-delay theory.)

The notch occurs because the amplitude spectrum of the surface temperature is flat — also found by Eschenbach, independently via different means — while the TSI amplitude spectrum has, of course, a pronounced peak around 11 years.

If low-altitude cloud cover troughed at every sunspot maxima (as suggested by Fig. 2-12 of Lockwood et. al.’s Earthshine Mission case from 2004), perhaps in response to troughs in galactic cosmic rays at sunspot maxima, then presumably albedo would also trough, and surface temperature peak, during sunspot maxima. This would make the empirical fact of the notch even *more* remarkable.

The notch is *not* intrinsic to the main part of the notch-delay hypothesis. The force ND hypothesis coming up allows for force D, the most consequential part of the hypothesis, which does not depend on either the notch’s existence (datasets are subject to revision) or its meaningfulness (under the notch-delay hypotheses below, the formal system is not quite invariant because the delay is one sunspot cycle and the duration of a sunspot cycle varies with time).

### Implications of the Notch for Climate Influences

The TSI peak every sunspot cycle fails to cause a corresponding peak in the Earth’s surface temperature record — therefore a countervailing cooling influence is present at precisely the times when TSI peaks.

The duration of the sunspot cycle varies considerably, from 9 to 14 years, yet notching implies that the countervailing influence is always synchronized to the TSI peaks — therefore the timing of the countervailing influence is controlled by the Sun.

The countervailing influence completely counters the warming influence of the TSI peaks — therefore the countervailing influence is as at least as strong as the direct heating effect of the changes in TSI.

### The Indirect Solar Sensitivity (ISS)

The second robust feature of the empirical transfer function is that it is remarkably flat for periods over 20 years (frequencies less than a twentieth of a cycle per year). For slower TSI fluctuations the sensitivity of surface warming to increases in TSI, herein called the **indirect solar sensitivity** (ISS), appears to be relatively constant. For periods over 200 years, the ISS is ~1.7±0.2 °C W^{−1} m^{2} (the value of the black line in Fig. 2 as the period increases). This is well above the direct solar sensitivity of 0.12 [0.07, 0.36] °C W^{−1} m^{2} in Eq. (1), suggesting that the long term influence of the TSI on ΔT_{S} is ~14 [4.2, 27] times larger than the direct heating effect of TSI.

Hence changes in TSI signal something that has an effect on the surface temperature that is much larger than the direct heating effect of changes in TSI — which is compatible with the finding in post 10 about the relatively large influence of EDA.