On the Earth, reasoning from experience, natural and climatic processes are significantly initiated and controlled by external forcing. It has a complex structure, but the Sun is a major contributor, acting directly and adjusting the other cosmic influences. Strengthening of the Sun's magnetic field is accompanied by an increase in the number of dark areas in the chromosphere, named sunspots. A linear combination of the sunspot numbers and sunspot groups has been chosen as a comprehensive indicator of solar activity. These combinations are called the Wolf numbers, and have no physical dimensions. Astronomers have been counting spots for about 400 years. As a result, a unique series of instrumental data about the cosmos has been obtained. See the classical treatises by Vitinsky (1973), Chizhevsky (1976), and Shurgin (1999).

In origin and due to inadequate measurements, natural and climatic processes consist of heterogeneous components, which are not always known. Besides, the basic values now studied were chosen long ago in past centuries to meet the challenges of technological development. For processes in nature, this traditional approach is not obvious. It should be expected that investigations will have a greater effect when they consider factors that are, in a sense, immanent to the system under study.

The development of options for component and factor analysis could help to make advances in this direction; see e.g. Lyubushin (2007). These approaches are universal but formal in character. Their efficiency is directly related to the anisotropy of the cloud of points representing the state of the system being studied in a multidimensional space. In addition, the results presented by the superposition of the original values are not always amenable to meaningful interpretation; this requires empirical regularities inherent in the system.

Distinctive features of the problem to be solved are taken into account more by different variants of informal classification, based on the optimization of objective functions of an empirical nature. In such approaches, the decomposition of the original set into subsets is performed. Each subset contains as strongly as possible related elements, and the relationship between elements of different subsets must be weaker. For example, Tartakovsky et al. (2015) performed this for climate classification by iteration of the phase of temperature series.

Based on observations, we can draw conclusions about the consistency of the Sun's forcing and certain climatic processes. It is well known that the cyclic motions in the solar system are manifested in the unceasing change of seasons, in daily warming and nocturnal cooling. These changes reflect a determinism that partially characterizes the climate for a certain time interval. There are other facts concerning possible determinism or the high correlation coefficients of the intensity of cosmic rays, the radio flux density, and series of Wolf numbers. We refer here to the publications by Alekseev et al. (2012), Budovyj et al. (2006), Buharov (1993), and Kuklin (1984).

Discussions are underway about how climate in general and the temperature particularly are sensitive to solar variability; see, e.g., Scafetta (2014). It should agree with this author in the sense that the progress can be expected in the way of sophisticated analysis of observational data, but in terms of some basic physical principles. The classic phenomenological approach involves the formalization of empirical data. Exaggerating the observed facts, we shall have to formulate the principle: “external forcing inherently initiates and synchronizes elementary processes in the geospheres”. Then we support this principle by a formal definition: “synchronicity of processes is manifested in the coincidence of their essential signs”, which it is necessary to define reasonably. Thus, the synchronism is selected as an essential factor in solar–terrestrial relations.

Below we present our theory and supporting algorithms to be applied for the synchronous analysis of series of average monthly Wolf numbers and the series of average monthly temperatures measured at 818 weather stations in the Northern Hemisphere of the Earth from 1955 to 2010.

The approach presented and the results of its application for climate problems and in the field of dendrochronology were partially reflected in the publications, e.g., of Tartakovsky (2015). It should be noted that the degree of synchronicity, as information about the transition of systems to a new state, was supported algorithmically and applied to solve geophysical problems as reported by Lyubushin (2007).

In our case, the novelty is that synchronicity was constructively defined, resulting in the decomposition of the original series. This computational procedure highlights the latent essence in the measured values and provides new information about the influence of the Sun on the temperature in the atmospheric surface layer, as is presented in this paper.

Let us assume that the series of experimental data xk,l relate one to one to natural processes under study, which are also real and with limited energy. Here, k is the discrete argument that takes N values at a given interval of observations, and l is the number of the series and the weather station.

The series can be supplemented and extended by continuity to the whole real axis periodically, using an even or odd manner. For such series there exist both the discrete direct and inverse Fourier transform: Xν,l=1N∑k=0N-1xk,lexp⁡(-i2πνk/N),xk,l=∑ν=0N-1Xν,lexp⁡(i2πνk/N). Here i denotes the imaginary unit, and ν is the discrete frequency. The Fourier coefficients Xν,l are not complex numbers for the assumed conditions of extending the series.

Possibly related to their common origin, the similarity of the series will be significantly reduced if, for different numbers l and the same frequency ν, the Fourier coefficients Xν,l take the opposite signs. Moreover, the larger the contribution that is made in the expansion by the basis functions, the greater is the violation of this genetic similarity. Conservation of the signs of the Fourier coefficients can be interpreted as a manifestation of the deterministic relationships between the series within certain limits, and the stochastic variability has an opportunity to be reflected in absolute values of these coefficients. Therefore, we define the signs of the Fourier coefficients as the essential signs, and we shall use them for implementing the synchronous analysis. Below, other reasons will be found for this choice.

For a pair of the series that characterizes the influence of the Sun and each series associated with the climatic process, we introduce the “components with coincident signs” (CS) and “components with non-coincident signs” (NS). We refer to the procedure for pairwise decomposition of the series on such latent components as “decomposition on the selected set”, hereinafter “s-decomposition”. In this case, we select the series that describes the solar activity for the entire planet. This Wolf number series sk has the same properties in relation to the Fourier transform as the series xk,l.

Let us obtain CS- and NS-components, x^k,l and x̃k,l as follows: x^k,l=∑ν=0N-1X^ν,lexp⁡(i2πνk/N),X^ν,l=Xν,l,signSν=signXν,l0,signSν≠signXν,l;x̃k,l=∑ν=0N-1X̃ν,lexp⁡(i2πνk/N),X̃ν,l=0,signSν=signXν,lXν,l,signSν≠signXν,l. The result of decomposition of the selected series sk, namely its CS- and NS-components (s^k,l and s̃k,l) depends on index l, which is individual for each series: s^k,l=∑ν=0N-1S^νexp⁡(i2πνk/N),S^ν,l=Sν,signSν=signXν,l0,signSν≠signXν,l;s̃k,l=∑ν=0N-1S̃νexp⁡(i2πνk/N),S̃ν,l=0,signSν=signXν,lSν,signSν≠signXν,l. These expressions describe the variable influence of solar activity across the Earth's surface.

Note that s-decomposition leaves unchanged the original values of the Fourier coefficients Xν,l and Sν, but each coefficient falls either in the CS- or in the NS-component of the series with index l.

We start from definitions (Eqs.  and ); they imply the validity of the equalities: xk,l=x^k,l+x̃k,l,Xν,l=X^ν,l+X̃ν,l;sk=s^k,l+s̃k,l,Sν=S^ν,l+S̃ν,l. Entering the scalar product and taking into account that it is invariant with respect to the Fourier transform, then using Eqs. ()–(), we obtain a set of useful expressions: ∑k= 0N-1sk⋅xk,l=(sk,xk,l)∝(Sν,Xν,l),(x^k,l,x̃k,l)=(s^k,l,s̃k,l)=(s^k,l,x̃k,l)=(s̃k,l,x^k,l)=0,(X^ν,l,X̃ν,l)=(S^ν,l,S̃ν,l)=(S^ν,l,X̃ν,l)=(S̃ν,l,X^ν,l)=0; and (sk,xk,l)=(s^k,l,x^k,l)+(s̃k,l,x̃k,l),(s^k,l,x^k,l)>0,(s̃k,l,x̃k,l)<0. In these expressions the summation is performed on the indices k or ν, and all the series are real by construction.

Thus, the CS- and NS-components of the original series are orthogonal, resulting in the additive property of the scalar product of these components; see expressions ().

Let us make sure of the property of extreme correlation for the identical components of the original series. We derive an equality relating the correlation coefficients rl=corr(sk,xk,l), r^l=corr(s^k,l,x^k,l), and r̃l=corr(s̃k,l,x̃k,l): rl=corrsk,xk,l=corrs^k,l+s̃k,l,x^k,l+x̃k,l=r^l⋅vars^k,l⋅varx^k,lvarsk⋅varxk,l+r̃l⋅vars̃k,l⋅varx̃k,lvarsk⋅varxk,l;r^l>0,r̃l<0; where the correlation coefficients and variances are calculated on the index k.

In view of expressions () and (), we find: rl>0⇒r^l>rlalternatively, if rl<0⇒r̃l>rl.

From these expressions, it follows that s-decomposition extracts from a pair of series xk,l and sk the components with extreme correlation for each index l: CS-components with positive correlation and NS-components with a negative one. These properties are also the reason for the choice of signs of the coefficients of the Fourier transform as the essential signs to describe the synchronicity.

Considering that the average value of a function is proportional to the value of its Fourier transform at zero frequency, we obtain the equalities for the average values: xk,l=x^k,l+x̃k,l,X0,l=X^0,l+X̃0,l;sk=s^k,l+s̃k,l,S0=S^0,l+S̃0,l; where the summation is performed on the index k.

Bearing in mind the even continuation of the series and that the Wolf numbers sk are always positive, we obtain the result that S0 is real and positive. In cases where X0,l>0, in accordance with expressions () we obtain: X^0,l=X0,l,X̃0,l= 0,S^0,l=S0,S̃0,l= 0; alternatively, if X0,l<0, then: X^0,l= 0,X̃0,l=X0,l,S^0,l= 0,S̃0,l=S0. It can be concluded that there are two sets of values of the index l. In one of these sets, the averages of the CS-components are equal to the average values of the original series, and the averages of the NS-components are equal to zero. In the other set of values of the index l, the CS- and NS-components are interchanged.

If the temperature scale has the zero value, the CS- and NS-components (x^k,l and x̃k,l) form on average the zones delimited by a zero-isotherm, where the weather stations with positive or negative average temperatures are located. There is one-to-one mapping of average temperatures and average values of their components. What is more, the CS- and NS-components of the Wolf numbers (s^k,l and s̃k,l) are constant within the designated zones, – either zero or non-zero.

Let us look at the normalized second initial moments for the original series and their CS- and NS-components. The initial moments do not include the centering operation, i.e., the average value is kept in series; it can have a physical sense as, e.g., a positive constant component of the solar activity. Naturally, the second initial moments are positive by definition.

Let the series x^k,l and s^k,l correspond to the moments ϑ^l and η^l; analogously, for the series x̃k,l and s̃k,l the moments ϑ̃l and η̃l are calculated, i.e., ϑ^l=(x^k,l,x^k,l)(xk,l,xk,l),ϑ̃l=(x̃k,l,x̃k,l)(xk,l,xk,l),η^l=(s^k,l,s^k,l)(sk,sk),η̃l=(s̃k,l,s̃k,l)(sk,sk). Taking into account the orthogonality and additivity – see expressions () and () – we find that: ϑ^l+ϑ̃l=1,η^l+η̃l=1. In addition, Pearson's correlation coefficient of these moments has the properties: r2=corrη^l,ϑ^l=corrη̃l,ϑ̃l=-corrη^l,ϑ̃l=-corrη̃l,ϑ^l.

We used the series of the average monthly temperatures for the calculations. These were measured at 818 weather stations of the Northern Hemisphere from 1955 to 2010. This is the most complete archive of global data on surface temperature in the public domain. The archive is updated regularly in the Met Office Hadley Centre observations datasets. For the same time interval, the series of the monthly average Wolf numbers were taken on site at the Pulkovo Observatory.

In order to consolidate homogeneous information, the original data were subdivided into monthly groups. The terms of the series for a particular month in each successive year were selected and inserted in the group without changing the original order. In total there were 12 temperature series for each of the 818 weather stations; and 12 series of Wolf numbers, which are identical at all weather stations. Each group includes 56 terms.

The discrete Fourier transform is the base algorithm of the s-decomposition. Scafetta (2014) rightly wrote about the inadmissibility of the formal application of the discrete Fourier transform and the subsequent misinterpretation of the results. Doing discrete spectral analysis, make sure that the Fourier coefficients decrease rapidly within a finite interval of definition. This problem had solved, e.g., in Tartakovsky (1993) by means of the preliminary polynomial filtering and by optimal continuation of series beyond the domain of definition.

In this case, to calculate the Fourier coefficients, the series were continued beyond the interval of definition, periodically in an even manner. Moreover, to check and eliminate the influence of the series length on the results, these series were interpolated from the original 56 up to 212 samples, having been performed on the index k, i.e., on the time. The finiteness in the Fourier transform of the continued series and the compliance of both the procedures of sampling and interpolation with the sampling theorem were monitored during calculations, which were realized by means of the Mathcad package.

The designations remain the same for the newly formed series: xk,l and sk. They will be referred to in the text below as the original series, unlike the series obtained by s-decomposition, i.e. the CS- and NS-components: x^k,l and s^k,l, x̃k,l and s̃k,l, respectively. The index k corresponds to the time, and l denotes the numbers of weather stations.

In this section, we analyze the foregoing reasoning and its consequences. First, we note that the properties mentioned in Sect. 3 were proved computationally with machine accuracy. This makes it possible to have confidence in the quality of the developed algorithms.

In the next step, the obtained algorithms were applied to the unique observational data. Initial moments, correlation coefficients, and mean values were calculated for these series. The computed dependencies were compared as far as possible with the climate geography and are discussed below.

This work is based, as a consequence of experience, on the hypothesis that synchronicity is an essential feature of solar–terrestrial relations. The Sun's external forcing initiates climatic processes on the Earth and should therefore be manifested in the similarity of their essential signs. Solar activity is characterized by the Wolf numbers and is considered as an integral indicator of forcing. Use is made of the orthogonal CS- and NS-components of Wolf numbers and the processes under study that differ in the coincidence and non-coincidence of their essential signs. These components are the latent essence of the phenomenon, and between them the information from the original data is redistributed without any distortion. To calculate these components, algorithms based on the Fourier transform were developed.

The theory is applied to decompose the Wolf numbers and temperature series from 818 weather stations of the Northern Hemisphere from 1955 to 2010. We obtained the following results.

The CS- or NS-components of the Wolf numbers and temperature series have significant correlation coefficients in the range from weak to strong values for small samples, typical for the minimum period of stability of the climate.

The histograms of the original temperature series coincide with the histograms of their components over long continuous ranges of temperatures, excluding the range of about ±3 ∘C.

The second initial moments of the CS-components of the Wolf numbers display the climate geography, and fall into two ranges, the width and the distance between which is about 30 % of the possible changes.

The relationship between the CS-components of the Wolf numbers and temperature series can be interpreted as an inflow of energy from the Sun, and between the NS-component – as an energy outflow. The distribution of the inflow and outflow of solar energy over the weather stations undergoes jumps from about 27 % in January to 39 % in May, excluding July and August.

This new approach is informative; it describes the manifestation of the forcing and corresponds to the known concepts of natural and climatic processes. It deserves wide application and the search for other matches or mismatches. The results are convincing that the things, “which are seen”∗, sometimes do not reflect the essence of the phenomenon; it may be helpful to look at latent things, “which are not seen”∗ and are novel at least.