# Reconstruction of the Interannual to Millennial Scale Patterns of the Global Surface Temperature

## Abstract

**:**

## 1. Introduction

## 2. Evaluation of the Confidence Levels of the Spectral Analysis

## 3. High-Resolution Spectral Analysis of the Global Surface Temperature Versus the CMIP5 Multi-Model Mean Function

## 4. Optimized Spectral Analysis and Harmonic Modeling

#### 4.1. The Non-Harmonic Temperature Component is Assumed to be Simulated by the Anthropogenic + Volcano CMIP5 Multi-Model Mean Temperature Function

#### 4.2. The Non-Harmonic Temperature Component is Assumed to be Simulated by 50% of the Anthropogenic + Volcano CMIP5 Multi-Model Mean Temperature Function

`mscohere.m`.

## 5. Secular and Millennial Temperature Reconstruction and a Discussion on the Physical Origin of the Climatic Oscillations

#### 5.1. The Optimized Semi-Empirical Climate Regression Model

#### 5.2. The Astronomically Optimized Semi-Empirical Climate Model

## 6. Validation of the Model Forecast

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**HadCRUT4 global surface temperature (red) (http://www.cru.uea.ac.uk/) [37] from 1850 to 2013 against the Coupled Model Intercomparison Project 5 (CMIP5) multi-model mean (blue) of historical plus the average among the RCP4.5, RCP6.0, RCP8.5 scenarios (http://climexp.knmi.nl) from 1861 to 2013. The green area is the 95% (=$2\sigma $) temperature confidence of the net (bias + measurement + sampling + coverage) error uncertainty. The blue curve is depicted shifted down for visual convenience.

**Figure 2.**(

**A**) $1\sigma $ global surface temperature confidence based on the net bias, measurement, sampling and coverage error uncertainty (yellow area) [37] and three computer-generated Gaussian noise records with a variable standard deviation modulated on the temperature uncertainty. (

**B**) Power spectra of the three computer-generated Gaussian noise records with their 95% and 99% average confidence level using the Multi-Taper Method (MTM) [118,119].

**Figure 3.**Power spectra of the global surface temperature record (

**A**) and the CMIP5 multi-model mean function (

**B**). The power spectra are calculated using the Multi-Taper Method (MTM) and the Maximum Entropy Method (MEM) using the SSA-MTM toolkit for spectral analysis [118,119]. The 95% and 99% confidence levels are deduced from the analysis of the theoretical temperature error analyzed in Figure 2.

**Figure 4.**The global surface temperature (blue) against the regression model made of Equation (1) + Equation (2) (red) using: (

**A**) The MTM spectral peak frequencies ($rmsr\approx 0.069$ °C); (

**B**) the MEM spectral peak frequencies ($rmsr\approx 0.068$ °C); (

**C**) the regression optimized frequencies are used ($rmsr\approx 0.061$ °C). See Table 1.

**Figure 5.**(Black) solar signature at the surface reproduced by the GISS ModelE from 1945 to 2003 [68,120]. (Red) reconstruction of the solar signature modeled by the global circulation models (GCMs) using the total solar irradiance record by Wang et al. [70]. (Blue) CMIP5 multi-model mean simulation (from Figure 1). (Purple) Coupled Model Intercomparison Project 5 (CMIP5) multi-model mean simulation detrended of the solar signature, Equation (3).

**Figure 7.**(

**A**) Extension of the CMIP5 multi-model mean simulation referring to the solar signature (red and black) and the greenhouse gas (GHG)+Aerosol+Volcano (blue and purple) signature. The extensions are constructed by calibrating the correspondent energy balance model outputs of (Crowley [125], Figure 3)on the CMIP5 multi-model mean estimates. (

**B**) Comparison between the extended CMIP5 multi-model mean simulation (cyan) made by summing the black and blue curved in A Equation (5) against the temperature proxy reconstruction by Moberg et al. [124] (red) calibrated and extended with the HadCRUT record (green) since 1850.

**Figure 8.**(

**A**) HadCRUT4 detrended of 50% of the CMIP5 multi-model mean simulation relative to anthropogenic and volcanic forcings alone (blue curve in Figure 5), Equation (6). (

**B**) Record depicted in A detrended of the secular and the millennial oscillations: see Equations (7) and (8). (

**C**) MTM and MEM power spectra of the record depicted in A.

**Figure 9.**(

**A**,

**C**,

**E**) show in blue the natural variability of the global surface temperature (Figure 8A, Equation (6)) versus the harmonic modeled using only the harmonic component of Equation (9), which also includes Equation (7), with the three parameter-sets (MTM, MEM and ROF) reported in Table 2. (

**B**,

**D**,

**F**) show in blue the HadCRUT4 global surface temperature record versus the full climate model made of the observed oscillations plus the anthropogenic and volcano effects modeled by Equation (9) with the parameters reported in Table 2. The $rmsr$ is between 0.06 and 0.07 °C.

**Figure 10.**Spectral coherence between HadCRUT4 global surface temperature-independent intervals (1861–1937 and 1937–2013). Coherent spectral peaks are observed at about 2.14, 2.85, 3.16, 3.5, 4.1, 4.8, 7.5, 9.3, 13.8 and ∼20 year. The covariance coherence analysis uses $L=52$ year moving windows.

**Figure 11.**(

**A**) HadCRUT4 global surface temperature record (red) versus the full semi-empirical climate model (blue) made of the regression optimized oscillations plus the anthropogenic and volcano effects modeled by Equation (9) with the parameters reported in Table 2 using the regression optimized frequencies versus the original CMIP5 multi-model mean function. (

**B**) The full semi-empirical climate model (blue) by Equation (11) using the same oscillations and the GHG+Aerosol+Volcano signature deduced by the energy balance model by Crowley [125] rescaled by a factor 3/4 to reduce the ECS from 2 °C to 1.5 °C for CO${}_{2}$ doubling.

**Figure 12.**The magnitude squared coherence (MSC, with windows length $L=102$ years, Equation (10)) between the temperature component filtered of the anthropogenic and volcano signal (Figure 8A) and the speed of the sun relative to the barycenter of the solar system [42]. The Capon’s approach known as the minimum variance distortion-less response (MVDR) method [136]. Major coherence harmonics are found at about 3, 7.5, 20 and 60 years. An extended coherence at 10–12 years, which corresponds to the solar cycle, is also observed.

**Figure 13.**Comparison of the time-frequency analyses between the speed of the sun relative to the center of mass of the solar system (

**left**) and of the HadCRUT global surface temperature record (

**right**). Details are found in Scafetta [106].

**Figure 14.**Astronomically optimized semi-empirical model (blue) uses the parameters listed in Table 3 using frequency and phases of the decadal to millennial harmonics deduced from astronomical considerations against a proxy temperature record (violet) and the instrumental temperature record (red). (

**A**) The anthropogenic plus volcano signature (black); the harmonic natural variability of the global surface temperature (green). The two curves are combined to obtain the blue curve. The $rmsr$ is about 0.07 °C. (

**B**) The GHG+Aerosol+Volcano signature (includes the anthropogenic signature since 1850) with $\lambda =1.5$ °C (black).

**Figure 15.**Comparison of the model (blue) against the latest global surface temperature HadCRUT4.6 (red), up to 2020.

**Table 1.**Regression coefficient of Equation (1) using the temperature MTM and MEM spectral peaks with a 99% confidence level depicted in Figure 3A and after a regression optimization of the frequencies. For a 153-year long sequence the spectral resolution is d$\Omega =1/153=0.0065$ yr${}^{-1}$ and the statistical error in the reported frequencies is $\nabla \Omega =\pm 0.5$ d$\Omega =\pm 0.003$ yr${}^{-1}$. The three sets of frequencies are compatible within their statistical error. ${\Omega}^{-1}$ is the period, $\Omega $ is the frequency, A is the amplitude of the sine wave and $\varphi $ is the phase.

MTM Spectral Peak Frequencies | MEM Spectral Peak Frequencies | Regression Optimized Frequencies | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\Omega}^{-\mathbf{1}}$($\mathrm{yr}$) | $\Omega $(${\mathrm{yr}}^{-\mathbf{1}}$) | $\mathbf{A}$(°C) | $\mathbf{\varphi}$ | ${\Omega}^{-\mathbf{1}}$($\mathrm{yr}$) | $\Omega $(${\mathrm{yr}}^{-\mathbf{1}}$) | $\mathbf{A}$(°C) | $\mathbf{\varphi}$ | ${\Omega}^{-\mathbf{1}}$($\mathrm{yr}$) | $\Omega $(${\mathrm{yr}}^{-\mathbf{1}}$) | $\mathbf{A}$(°C) | $\mathbf{\varphi}$ |

62.11 | 0.0161 | 0.118 | 0.20 | 64.10 | 0.0156 | 0.122 | 0.16 | 65.79 | 0.0152 | 0.121 | 0.14 |

21.33 | 0.0469 | 0.042 | 1.00 | 20.75 | 0.0482 | 0.044 | 0.08 | 20.70 | 0.0483 | 0.046 | 0.10 |

10.24 | 0.0977 | 0.027 | 0.13 | 10.44 | 0.0958 | 0.025 | 0.97 | 10.21 | 0.0979 | 0.031 | 0.13 |

9.225 | 0.1084 | 0.038 | 0.45 | 9.234 | 0.1083 | 0.041 | 0.44 | 9.183 | 0.1089 | 0.039 | 0.50 |

8.190 | 0.1221 | 0.024 | 0.82 | 7.831 | 0.1277 | 0.015 | 0.38 | 8.190 | 0.1221 | 0.021 | 0.80 |

7.530 | 0.1328 | 0.024 | 0.66 | 7.645 | 0.1308 | 0.024 | 0.57 | ||||

6.131 | 0.1631 | 0.021 | 0.27 | 6.020 | 0.1661 | 0.027 | 0.61 | 6.254 | 0.1599 | 0.027 | 0.94 |

5.277 | 0.1895 | 0.025 | 0.33 | 5.200 | 0.1923 | 0.025 | 0.49 | 5.236 | 0.1910 | 0.029 | 0.42 |

4.762 | 0.2100 | 0.031 | 0.04 | 4.746 | 0.2107 | 0.033 | 0.10 | 4.773 | 0.2095 | 0.034 | 0.01 |

4.232 | 0.2363 | 0.017 | 0.46 | 4.202 | 0.2380 | 0.022 | 0.58 | 4.179 | 0.2393 | 0.026 | 0.65 |

3.644 | 0.2744 | 0.025 | 0.70 | 3.635 | 0.2751 | 0.024 | 0.73 | 3.628 | 0.2756 | 0.029 | 0.79 |

3.531 | 0.2832 | 0.023 | 0.89 | 3.516 | 0.2844 | 0.021 | 0.95 | 3.556 | 0.2812 | 0.028 | 0.79 |

2.876 | 0.3477 | 0.025 | 0.95 | 2.869 | 0.3486 | 0.025 | 0.02 | 2.875 | 0.3478 | 0.025 | 0.97 |

**Table 2.**Regression coefficient of Equation (1) using the temperature MTM and MEM spectral peaks with a 99% confidence level depicted in Figure 8C and after a non-linear regression optimization of the frequencies. The three sets of frequencies are compatible within the spectral resolution of the analysis. ${\Omega}^{-1}$ is the period, $\Omega $ is the frequency, A is the amplitude of the sine wave and $\varphi $ is the phase. The $rmsr$ is 0.06–0.07 °C.

MTM Spectral Peak Frequencies | MEM Spectral Peak Frequencies | Regression Optimized Frequencies | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\Omega}^{-\mathbf{1}}$($\mathrm{yr}$) | $\Omega $(${\mathrm{yr}}^{-\mathbf{1}}$) | $\mathbf{A}$(°C) | $\mathbf{\varphi}$ | ${\Omega}^{-\mathbf{1}}$($\mathrm{yr}$) | $\Omega $(${\mathrm{yr}}^{-\mathbf{1}}$) | $\mathbf{A}$(°C) | $\mathbf{\varphi}$ | ${\Omega}^{-\mathbf{1}}$($\mathrm{yr}$) | $\Omega $(${\mathrm{yr}}^{-\mathbf{1}}$) | $\mathbf{A}$(°C) | $\mathbf{\varphi}$ |

760 | 1/760 | 0.350 | 0.171 | 760 | 1/760 | 0.350 | 0.171 | 760 | 1/760 | 0.33 | 0.171 |

115 | 1/115 | 0.050 | 0.424 | 115 | 1/115 | 0.050 | 0.424 | 115 | 1/115 | 0.055 | 0.424 |

58.50 | 0.0171 | 0.086 | 0.26 | 58.82 | 0.0170 | 0.088 | 0.25 | 63.29 | 0.0158 | 0.091 | 0.17 |

21.32 | 0.0469 | 0.029 | 0.94 | 20.33 | 0.0492 | 0.033 | 0.08 | 20.33 | 0.0492 | 0.034 | 0.10 |

10.34 | 0.0967 | 0.023 | 0.024 | 10.72 | 0.0933 | 0.012 | 0.85 | 10.37 | 0.0964 | 0.025 | 0.00 |

9.225 | 0.1084 | 0.032 | 0.46 | 9.302 | 0.1075 | 0.033 | 0.37 | 9.19 | 0.1088 | 0.034 | 0.49 |

7.587 | 0.1318 | 0.020 | 0.61 | 7.680 | 0.1302 | 0.020 | 0.52 | 7.536 | 0.1327 | 0.020 | 0.65 |

6.131 | 0.1631 | 0.024 | 0.25 | 6.039 | 0.1656 | 0.027 | 0.51 | 6.010 | 0.1664 | 0.029 | 0.61 |

5.252 | 0.1904 | 0.026 | 0.39 | 5.198 | 0.1924 | 0.024 | 0.51 | 5.219 | 0.1916 | 0.026 | 0.44 |

4.785 | 0.2090 | 0.031 | 0.96 | 4.762 | 0.2100 | 0.031 | 0.03 | 4.787 | 0.2089 | 0.033 | 0.95 |

3.644 | 0.2744 | 0.024 | 0.70 | 3.640 | 0.2747 | 0.024 | 0.72 | 3.636 | 0.2750 | 0.027 | 0.75 |

3.531 | 0.2832 | 0.023 | 0.91 | 3.516 | 0.2844 | 0.021 | 0.96 | 3.554 | 0.2814 | 0.025 | 0.81 |

2.876 | 0.3477 | 0.025 | 0.96 | 2.871 | 0.3483 | 0.026 | 1.00 | 2.874 | 0.3480 | 0.025 | 0.98 |

**Table 3.**The harmonic parameters of the astronomically optimized semi-empirical model. Here the 6 frequency and phases of the decadal to millennial harmonics were substituted with the theoretical values deduced from astronomical considerations. About the millennial cycle note that 760 year is not its period, see the full equation is Equation (7) and the explanation in the text. For the probable physical origin of the oscillations see Refs. [84,85,86,106].

Semi-Empirical Optimized Frequencies | ||||
---|---|---|---|---|

Possible Origin | ${\Omega}^{-\mathbf{1}}$($\mathrm{yr}$) | $\Omega $(${\mathrm{yr}}^{-\mathbf{1}}$) | $\mathbf{A}$(°C) | $\mathbf{\varphi}$ |

Solar/Planetary | 760 | 760${}^{-1}$ | 0.3228 | 0.1711 |

Solar/Planetary | 115 | 115${}^{-1}$ | 0.0585 | 0.4239 |

Solar/Planetary | 61 | 61${}^{-1}$ | 0.0859 | 0.152 |

Solar/Planetary | 20 | 20${}^{-1}$ | 0.0334 | 0.148 |

Sun Spots | 10.4 | 10.4${}^{-1}$ | 0.0241 | 0.020 |

Solar-Lunar tidal | 9.3 | 9.3${}^{-1}$ | 0.0265 | 0.497 |

Solar/Planetary | 7.463 | 0.1340 | 0.0216 | 0.711 |

Solar/Planetary | 6.003 | 0.1666 | 0.0272 | 0.617 |

Solar/Planetary | 5.238 | 0.1909 | 0.0260 | 0.409 |

Solar/Orbital | 4.795 | 0.2086 | 0.0326 | 0.931 |

Solar/Orbital | 3.633 | 0.2752 | 0.0276 | 0.767 |

Solar/Orbital | 3.556 | 0.2812 | 0.0254 | 0.792 |

Solar/Orbital | 2.874 | 0.3480 | 0.0247 | 0.975 |

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Scafetta, N. Reconstruction of the Interannual to Millennial Scale Patterns of the Global Surface Temperature. *Atmosphere* **2021**, *12*, 147.
https://doi.org/10.3390/atmos12020147

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Scafetta N. Reconstruction of the Interannual to Millennial Scale Patterns of the Global Surface Temperature. *Atmosphere*. 2021; 12(2):147.
https://doi.org/10.3390/atmos12020147

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Scafetta, Nicola. 2021. "Reconstruction of the Interannual to Millennial Scale Patterns of the Global Surface Temperature" *Atmosphere* 12, no. 2: 147.
https://doi.org/10.3390/atmos12020147