What this research found
The Mauna Loa carbon dioxide record — 408 monthly observations from January 1990 through December 2023 — was separated into trend, seasonal, and residual components, then used to fit a seasonal ARIMA model whose 24-month forecast was scored against observations held back entirely from fitting. The seasonal cycle averages 6.55 parts per million peak to trough and reaches its maximum in May in all 34 years. The annual growth rate accelerated from 1.57 parts per million per year in the 1990s to 2.47 in 2014 to 2023, and the two-year-ahead forecast landed within a root-mean-square error of 0.988 parts per million.
- The seasonal cycle is strikingly regular: mean peak-to-trough amplitude of 6.55 parts per million with a standard deviation of 0.23 across 34 years. The annual maximum falls in May in every single year, and the minimum in October in 20 years and September in the other 14.
- Growth accelerated by 57%. The mean annual rise went from 1.57 parts per million per year in 1990 to 1999 up to 2.47 in 2014 to 2023, an absolute increase of 0.90, and three independent estimation methods landed within 0.06 of each other in each decade.
- The seasonal amplitude itself shows only a weak upward trend of 0.0065 parts per million per year, which is not statistically significant over this window (p = 0.10, R-squared 0.08). The report attributes this to Mauna Loa's subtropical location, since the documented amplification of the seasonal cycle is strongest at high northern latitudes.
- An exhaustive search across 36 specifications selected a SARIMA(1,1,1) by (0,1,1) model at period 12 with an Akaike information criterion of 242.90, ahead of its nearest competitors at 243.31 and 243.91. Its residuals are indistinguishable from white noise, with Ljung-Box p-values of 0.92, 0.67 and 0.73 at lags 12, 24 and 36.
- The held-out 24-month forecast reached a root-mean-square error of 0.988 parts per million, a mean absolute error of 0.933, and a mean absolute percentage error of 0.22%, with 22 of 24 observations inside the 95% prediction intervals. It under-predicted in 23 of 24 months by a mean of 0.93, consistent with the unusually fast growth of 2024 that a model calibrated on earlier decades could not anticipate.
How it was done
The full NOAA monthly-mean record was retrieved and split into a 408-month analysis window covering January 1990 to December 2023 and a completely held-out 24-month validation window for 2024 and 2025, both verified as contiguous and free of gaps. The gap-filled monthly-mean series was used as the input rather than NOAA's own deseasonalised column, which would have presupposed the very decomposition being performed. Seasonal-trend decomposition using Loess separated the additive trend, seasonal, and residual components with robust fitting, and per-year peak-to-trough amplitudes and modal peak and trough months were then extracted. Decadal growth rates were computed three ways — annual differences of the trend, twelve-month differences of the raw series, and an ordinary least squares slope on the trend — to guard against method-specific artefacts. A grid search over 36 seasonal ARIMA order combinations, with non-seasonal and seasonal differencing fixed in advance, selected the model that was then diagnosed and used to forecast 24 months ahead with prediction intervals.
Data sources
- NOAA Global Monitoring Laboratory Mauna Loa monthly-mean CO2 record — 820 monthly values from March 1958 to June 2026, file vintage 5 July 2026, reported as dry-air mole fractions on the WMO X2019 calibration scale
- Cleveland et al., Journal of Official Statistics 6(1):3 (1990) — the seasonal-trend decomposition using Loess procedure
- Graven et al., Science 341:1085 (2013) and Forkel et al., Science 351:696 (2016) — documented amplification of the northern seasonal CO2 exchange
- Betts et al., Nature Climate Change 6:806 (2016) and Global Carbon Budget 2023 — context for El Niño-amplified growth-rate anomalies
Limitations
Mauna Loa is a single subtropical station sampling background mid-Pacific air, so its seasonal amplitude is smaller than at high northern-latitude sites and the weak amplitude trend found here is not evidence against biospheric amplification elsewhere. A seasonal ARIMA is univariate and linear and therefore cannot anticipate exogenous shocks such as El Niño forcing or emissions changes, and the temporary relocation of measurements to Maunakea after the 2022 Mauna Loa eruption introduces a minor inhomogeneity near the end of the fitting window.
How this research was produced
K-Dense Web planned and ran this environment investigation end to end — gathering the sources, carrying out the analysis, producing the figures, and drafting the report. The full session transcript, including every intermediate step, is available to view.
