Model. T(t) = a + b·Niño3.4(t − Le) + c·AOD(t − Lv) + d·TSI(t − Ls) + e·5.35 ln(CO₂(t)/CO₂(1959)). Temperature: monthly GISTEMP v4; El Niño: Niño 3.4 (NOAA PSL); volcanoes: stratospheric aerosol optical depth at 550 nm (GISS, to September 2012); Sun: NRLTSI2 to 2017 then TSIS-1, shifted by their mean difference over 2018–2023; human: radiative forcing of Mauna Loa CO₂ (Myhre et al.), standing for all human influences, gases and aerosols, in their proportion over the period.
Fit. Ordinary least squares from January 1959 to September 2012, the only period when the four factors are measured. The lags of the natural factors are chosen by the best fit on a grid (El Niño 0–6 months, volcanoes 0–12, Sun 0–2), like Foster and Rahmstorf (2011). Standard errors are corrected for the autocorrelation of the residuals by the factor √((1 + φ)/(1 − φ)) (Santer et al., 2000).
Validation. The frozen coefficients predict the following months, never seen by the fit, with aerosols at their background level (median 2000–2012): a real test of the model.
Adjusted trend. Linear regression of the adjusted temperature since 1979, same autocorrelation correction.
- GISTEMP v4 global land-ocean temperature anomaly
-
GISTEMP Team (2026). GISS Surface Temperature Analysis (GISTEMP), version 4. NASA Goddard Institute for Space Studies; Lenssen et al. (2024).
· doi:10.1029/2023JD040179 · source page ↗
Licence : NASA data, public domain; cite GISTEMP Team and Lenssen et al. (2024) · retrieved on Oct 23, 2026 · SHA-256 4c64eda288bc… - Niño 3.4 SST anomaly (NOAA PSL, ERSSTv5)
-
NOAA PSL, Niño 3.4 index: monthly SST anomaly (5°N–5°S, 170°–120°W) from ERSSTv5, climatology 1991–2020.
· source page ↗
Licence : U.S. Government work, public domain · retrieved on Oct 23, 2026 · SHA-256 879cf3cb31d7… - NASA GISS stratospheric aerosol optical depth (Sato et al., updated to 2012)
-
Sato, M., Hansen, J. E., McCormick, M. P., & Pollack, J. B. (1993). Stratospheric aerosol optical depths, 1850–1990. J. Geophys. Res., 98, 22987–22994; updated through 2012 by NASA GISS.
· doi:10.1029/93JD02553 · source page ↗
Licence : NASA data, public domain; cite Sato et al. (1993) · retrieved on Oct 23, 2026 · SHA-256 40b245c8fc87… - NRLTSI2 daily total solar irradiance (NOAA CDR via LASP LISIRD)
-
Coddington, O., Lean, J. L., Pilewskie, P., Snow, M., & Lindholm, D. (2016). A Solar Irradiance Climate Data Record. Bulletin of the American Meteorological Society, 97, 1265–1282. Data: NRLTSI2 via LASP LISIRD.
· doi:10.1175/BAMS-D-14-00265.1 · source page ↗
Licence : NOAA Climate Data Record, public domain; cite Coddington et al. (2016) · retrieved on Oct 23, 2026 · SHA-256 f6ca2f1a23d4… - TSIS-1 TIM daily total solar irradiance (LASP LISIRD)
-
Kopp, G., et al. TSIS-1 Total Irradiance Monitor (TIM) daily-averaged total solar irradiance at 1 au, Level 3, LASP LISIRD (tsis_tsi_24hr).
· source page ↗
Licence : NASA/LASP data, freely available; cite Kopp et al. and the TSIS-1 TIM team - Mauna Loa monthly mean CO₂ (NOAA GML)
-
Lan, X., Tans, P., & Thoning, K. W. (2026). Trends in globally-averaged CO₂ determined from NOAA Global Monitoring Laboratory measurements; Mauna Loa monthly mean, after Thoning et al. (1989).
· doi:10.1029/JD094iD06p08549 · source page ↗
Licence : U.S. Government work, public domain; cite Lan, X., Tans, P. & Thoning, K. W., NOAA GML · retrieved on Oct 23, 2026 · SHA-256 d895fb5b4a59… - Lean et al. (2008)
-
Lean, J. L., & Rind, D. H. (2008). How natural and anthropogenic influences alter global and regional surface temperatures: 1889 to 2006. Geophysical Research Letters, 35, L18701.
· doi:10.1029/2008GL034864
Licence : Published literature; cited - Foster et al. (2011)
-
Foster, G., & Rahmstorf, S. (2011). Global temperature evolution 1979–2010. Environmental Research Letters, 6, 044022.
· doi:10.1088/1748-9326/6/4/044022
Licence : Published literature; cited - Santer et al. (2000)
-
Santer, B. D., Wigley, T. M. L., Boyle, J. S., Gaffen, D. J., et al. (2000). Statistical significance of trends and trend differences in layer-average atmospheric temperature time series. Journal of Geophysical Research, 105(D6), 7337–7356.
· doi:10.1029/1999JD901105
Licence : Published literature; cited - Robock (2000)
-
Robock, A. (2000). Volcanic eruptions and climate. Reviews of Geophysics, 38(2), 191–219.
· doi:10.1029/1998RG000054
Licence : Published literature; cited - Goessling et al. (2025)
-
Goessling, H. F., Rackow, T., & Jung, T. (2025). Recent global temperature surge intensified by record-low planetary albedo. Science, 387(6729), 68–73.
· doi:10.1126/science.adq7280
Licence : Published literature; cited