One blue line

Each of the past five years, the warmest in human history and years inclusive of warm and cooler phases of the earth’s natural weather cycle, has been important settling the climate change debate once and for all.  

If the solid blue curved line “fits” the dots in this data, that’s really all anybody needs to know about the climate change discussion: if the solid blue curved line fits, no debate, we are all agreed, including the Trump Administration. No need for talk, no complex models, forecasts of the future, no further study needed, no more book tours – if it fits, this climate change matter is fully settled.

The solid blue curve line clearly fits this data and, therefore, climate change is now fully settled. The last ten years of data have settled the matter: all in agreement, “immediate, aggressive” action to mitigate is “imperative.”

Figure 1: 10-year moving average of the global temperature curve fit with a blue line

Now I defend my claim. All math can easily be repeated with Excel and is generally high school-level math. Whereas “Math is linked in the popular mind with phobia and anxiety. You’d think we’re discussing spiders.” (Strogatz,2014, page 281), there are two math concepts used: 1. the dots on the graph are the observed global temperatures in a specific period of time, where the period across the bottom line (x-axis) is a year and the dot above that year is the temperature observed that year. This forms the temperature dataset: for year 1, the dot for that year, and so on. Year 1 is 1880, year 146 is 2025, for example. The other concept is how quickly this data is changing – is it growing and if so by how much, is the data growing for one amount for a bit, and then a higher rate later, for example. 

The graphs and figures tell the story visually as well.  I have learned over the years of talking about climate change, even very smart, well educated people are not necessarily conversant in the confusing language of math.

Actual global temperatures over the past 10 years, including 2023, 24, and 25, are breakout years and are the warmest in the past 11,700 years¹. 

Figure 2: Graphs of global temperatures showing the range of the prior 100,000 years and global temperature relative to natural variation²

Most people know: the climate has already changed; the effects of climate change are evident in their daily lives. 

Let’s start with what is agreed. The Trump Administration’s Department of Energy established a threshold for action. The Department of Energy convened a “Climate Working Group” of five scientists with a background in climate change. This Group produced a Draft Report in July 2025, A Critical Review of the Impacts of Greenhouse Gas Emissions on the U.S. Climate³. This Report has distilled the very complex conversation around climate change to one question: how rapidly is our climate changing?  Above a threshold, the DOE Report finds that “immediate aggressive emission controls become more imperative…” (Report Section 4.1). 

The solid curved blue line in Figure 1 is well above the DOE Report threshold - based the historical data alone – no supercomputers, no models of the climate, the DOE fully settles the matter, and “immediate aggressive” action is “imperative.” 

Figure 1 is the 10-year moving average of global temperature anomaly annual mean using the NASA GLB data set with a baseline of the “pre-industrial⁴.” Each dot is the prior 10 year period’s average annual global temperature. So, the first dot, which is plotted above year 1, which is 1890, is the average of each of the years of 1880 to 1890, the second dot plotted at year 2, which is 1891, the average of 1881 to 1891, and so on. See Figure 2 for each year’s global temperature.

The solid blue curved line is a mathematical equation “curve fit” to the data using the MATLAB programing language⁵. Feel free to ask Grok how MATLAB computes this or just look visually – this solid blue curved line clearly fits this data.

Two conclusions: (1) obviously, the sold blue curve does fit data fits the data; the “goodness of fit” is 0.95, with 1.0 being a perfect fit, and (2) this curve is “exponentially” growing. Exponential growth is well known, and indicates systems, in this case the earth’s temperature, that are growing in one period and then growing faster in the next period, and so on.

For those who want to know the math, the solid blue curved line’s exponential curve has a formula, Global Temperature in a period = 0.15 e^0.0303t, where “t” is the period in years from 1880, with 2025 being the 146th period, for example.  

Recall, we are trying to answer how rapidly is the global temperature is growing. This is a problem for High School math and not at all controversial or difficult. The rate of change of global temperature at a period =303 × 0.153 e^0.303t for t = 146 (the year 2025) is equal to 0.387 degrees C per decade. The underlying time series is the ten-year moving average, so this calculation is backward-looking. 

We find that the backward rate of change over the last twenty years is 0.303 degrees C per decade. This describes what the global temperature rate of change WAS in the past. The temperature in the past was growing more slowly than the global temperature is growing now – this rate of change is increasing or accelerating.

Using this 10-year moving average data rate of change, the curve fit equation yields a 2036 calculation of 0.524 degrees C per decade rate of growth. In MATLAB, I also used a standard time series model called ARIMA⁶ and calculated that the expected rate of growth in the next ten years is 0.51 degrees C per decade, see Figure 2.

Figure 3: ARIMA forecast of the next ten years of climate change growth

By way of reference, the average rate of change over the entire 1880 to 2025 period is about 0.2 degrees C per decade and is discussed in the context of a “climate sensitivity” of 3.0 degrees C per doubling of CO₂. I stay away from using this climate sensitivity metric. While frequently quoted and important, the climate sensitivity is beyond the scope of this analysis.   

We know that the average temperature change over the full 146-year period is around 0.20 degrees C per decade, that the actual change over the past 20 years is around 0.30 degrees C per decade, the rate of change over the last 10 years is around 0.39 degrees C per decade (calculated both analytically by the curve fit equation, and numerically from the data itself, both have some result), and that two models of the 10-year moving average temperature data (the simple curve fit equation and the more rigorous ARIMA model have this rate of change growing to 0.51 and 0.52 degrees C per decade.  Furthermore, we also know that the global temperature data is growing exponentially with remarkable confidence.

These rates of growth metrics are clearly more than the DOE threshold for “immediate aggressive” action⁷.

There is no question that the climate change issue can be very complicated, and there are many unknowns, but the climate has already changed, and is changing very quickly, and this change is obvious to all, as is the fit of the solid curved blue line to the data in Figure 1. As to the question of whether governments should act, the earth has spoken very clearly; the Trump DOE threshold for action has clearly been met.

A few items for those who want to dig deeper. It is clear and obvious that any of these global temperature rate of change implies a climate sensitivity that exceeds the threshold of 4.5 degrees C set in the DOE Report^8,9.

I will look at the non-moving-averaged data briefly. Figure 4 is the full 146-year record set that is not averaged; that is, every dot is the global temperature for the period, with the first year being 1880 and the last year being 2025. Again, the solid blue curved line is a best fit for the data; again, this curve has an outstanding goodness of fit of 0.91, with 1.00 being perfect. And again, we clearly see the global temperature record growing exponentially at a rate of change of 0.404 degrees C per decade.

Figure 4: Global annual temperature anomaly in degrees C

Figure 5: Global Temperature anomaly with linear fit line

I hear all the time, “the average rate of change since 1880 is 0.18 (or 0.20) degrees C per decade (and the climate sensitivity is 3.0 degrees) no need to worry …”. This is true if the data was a straight line – it is not; and this assumption of linearity, a common “flaw of averages” is misinformation.

I hear, “its weather, it was hot, but will go back to being cold …” I really don’t hear this from experts by the way; but just not so – the temperature data is not “normally distributed” and we are not seeing natural variation, all tests for “normality” fail catastrophically, we are seeing the opposite in fact, not natural variation rather exponential growth.

I hear, “it must be the data…” Not so. There are various global temperature records available. A US organization called Berkley Earth keeps its own unbiased data set. I used the NASA dataset. I chose the NASA dataset because it is both publicly available and there is a peer reviewed published methodology. See Figure 6, which shows various datasets; the choice of dataset is not detectable in the outcome, nor does “urban island effect” or any other data distraction. See Dressler and Kopp 2025 for a complete discussion on data – the data discussion has been resolved for over a decade.

As a sanity check to the conclusion that the globe is warming, at an alarming rate that exceeds thresholds for immediate aggressive action, I refer to a very technical metric of how bright the earth is – a bright earth is irradiating heat back into space, a dark earth, the heat is trapped by greenhouse gases and not escaping. Sure enough, the heat is being trapped, which is why we are seeing the global temperature rise so quickly.

Figure 7: Brightness of Earth

The DOE Report was not well received by the scientific community; nevertheless, the establishment of a publicly stated threshold for action, a threshold the Earth has now clearly breached, hopefully will force the same public consensus that exists in the scientific community.

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References

1. James E. Hansen, et. al. (2025) Global Warming Has Accelerated: Are the United Nations and the Public Well-Informed? Environment: Science and Policy for Sustainable Development

2. Figure SPM.1, Sixth Assessment Report, IPCC, 2022

3. Koonin, et. al., A Critical Review of the Impacts of Greenhouse Gas Emissions on the U.S. Climate, Draft Report, US Department of Energy, July 2025 (“the DOE Report”)

4. NASA GISTEMPv4 Observational Uncertainty Ensemble, Nathan Lenssen (2024), Gavin A. Schmidt, Michael Hendrickson, Peter Jacobs, Matthew J. Menne, Reto Ruedy. Whereas the GLB data uses a baseline of the 1950 to 1980 average, I have adjusted to a baseline of 1890 to 1900 which is a “pre-industrial” baseline.

5. MATLAB (short for Matrix Laboratory) is a multi-paradigm programming language and interactive environment developed by MathWorks, a company founded in 1984. 

6. ARIMA stands for Autoregressive Integrated Moving Average. It's a widely used statistical model for analyzing and forecasting time series data, incorporating autoregression (AR), differencing to achieve stationarity (I), and moving average (MA) components.

7. Dessler, A.E. and R.E. Kopp (Ed.). (2025). Climate Experts’ Review of the DOE Climate Working Group Report

8. Hanson 2025

9. Dessler, A.E. and R.E. Kopp (Ed.). (2025).