by Dr. Alan Welch FBIS FRAS
Summary
This study started after considering the following question:-
“Satellite measurement of sea levels, release of data and analysis generally started in 1993. How would different models react to the analyses starting at later dates?”
A range of start dates were investigated. The main conclusions were that the concept that the sea level rises were accelerating and that the rate of acceleration is basically stable is shown to be untenable.
Having presented the different start date analyses and shown their findings, an appendix brings together all the studies showing that sea level rises follow a sinusoidal pattern. Hopefully this will reach a wider audience than WUWT and illicit a response to the work from other workers in this field.
Changing Starting Dates
Much of the concerns regarding sea level rises has arisen following the publication of two papers by Nerem et al (Ref 1 and 2) in 2018 and 2022 respectively. In these papers they use quadratic curve fitting to derive a perceived acceleration and then extrapolate into the future. Their two main conclusions are that the sea level rise could be modelled as a quadratic curve and that the “acceleration”, derived as 2 times the quadratic coefficient, is stable after 2017. These papers are highly cited and still appear in many Google searches regarding sea level rises.
When analysing data using a quadratic curve the outcome can be greatly influenced by the start date of the study and by the length of the data set. In my previous work (such as ref 3) a sinusoidal curve variation has been considered more appropriate, that variation being about the best linear fit. In the very long term, such as with Tidal Gauge data, periods of analysis may show other long term slower changes which can be ignored in satellite data analysis.
Using August 2026 data two studies were undertaken (see notes on data sources). Firstly, analysing the actual data and secondly analysing a sinusoidal best fit. A common feature, namely the change with time of the perceived “acceleration”, was used in judging effects this being derived from fitting a quadratic curve. Using this “acceleration” does not attach a physical meaning but the way it does or does not change with time can be very revealing.
The actual data involve some short-term variations due to the annual and semi-annual signal and many random short-term variations. These had a bigger effect when the actual data are analysed compared with a sinusoidal fit curve. Analysing short NASA data sets using less than 20 years of data could produce large fluctuations in “acceleration”.
The effect of the start date can be more fully studied for the sinusoidal curve as it can be extended backwards and forwards to give a more complete investigation. Pseudo data were generated for a 27.6-year period curve with an amplitude of +/- 3.8 mm.
The “accelerations” were calculated for several starting dates from 1963 to 2007 in steps of 4 years. A selection of the more extreme results was plotted as shown in Fig 1. The plots of “acceleration” are for start dates of 1963, 1971, 1979, 1987,1999 and 2007 and being based on a sinusoidal variation produce smooth curves right back to the start date. All of these curves show a decay in “acceleration” over time reaching near zero values after about 50 years.
Also shown are the “accelerations” calculated using the NASA data and part of the1993 start curve. This 1993 start graph has appeared in many reports (Ref 3, 6, and 7) on how sea levels have been changing and has been flagged as being one of the more important graphical presentations in judging various models. It is only shown from about 2020 onwards as prior to this the curve based on the NASA data becomes more erratic when shorter time periods are used.
Figure 1
These were sufficient to show the expected repeating pattern on roughly a 28-year cycle. There were near positive peaks for starts of 1979 and 2007 of about 0.2 mm/year2 and mirror image intermediate negative dips.
Turning to the actual data it is not possible to go back before 1993 and if a start date is too far on from 1993 the range of data may be too short to be meaningful.
A 1999 start date was chosen and the plot of “accelerations” is shown in figure 2 for the actual and sinusoidal data. It has been found that it requires nearly 20 years for the graph to settle. Up to 2006 the data used readings from the Topex satellite which seemed to be less reliable when compared with the Jason 1 satellite launched in 2001. The individual
satellite readings were available on a NOAA site, but this seems to have vanished. Data from NASA and University of Colorado also changes occasionally with no explanations provided. In extreme cases EVERY data value changed. A fuller description of the data sources is provided at the end of this paper. Only the latest NASA data are available and since 2018 the University of Colorado data have also been removed except for the latest.
Figure 2
It was not possible to carry out an analysis for the first dip as this would require an approximate start date of 2014 leaving only about 12 years of data.
The 1993 start analyses has been reported many times in monthly studies and yearly reports but the 1999 start results shows a higher peak in “acceleration”. In figure 2 the two curves show a similar higher peak of about 0.13mm/year2 although the build-up is much faster for the actual data possibly due to the fact that the data starts from a low base soon after 1999 which could result in much higher “decelerations” early on.
A copy of the 1993 start date is shown in figure 3.
Figure 3
As a start date of 2014 is not possible due to the short extent of the data a start date of 2007 was considered a possibility. Figure 4 shows the results. The “accelerations” now reach a sizeable negative “deceleration” stage with the 2 curves matching well over the 2019 to 2026 period. The satellites now used are Jason 1 (2001-2013), Jason 2 (2008 –
2019), Jason 3 (2016 -) and Sentinel-6 (2020-) which showed much better correlation where they overlapped.
Figure 4
The conclusions from these analyses are that with different starting dates the same data can produce as much “deceleration” as “acceleration” reaching values in the region of +/- 0.2 mm/year2.
Figures 2, 3 and 4 show starting dates of 1999, 1993 and 2007 and all show an agreement between a best fit sinusoidal set of data and the actual data although when the sinusoidal data can be analysed back to the start date the actual data analyses only settle down after a period of about 20 years. Over the years the 1993 start has been extensively studied, and it may be informative to extend this ongoing analysis to the 1999 and 2007 start dates.
The term “acceleration” was introduced in the Nerem et al paper of 2018. This was quickly picked up by the media and public and by using extreme extrapolation, initially up to 2100 but later up to 2050, completely wrong conclusions were reached concerning long-term sea-level rises. If it had been referred to as a quadratic coefficient it would be seen to be just what it is, a mathematical term. Also, the various figures show there is a wide range of “accelerations” from -0.2 to +0.2 mm/year2 depending on the start date of the analyses. This totally negates the Nerem et al 2022 statement that accelerations are stable after 2017.
With these new findings I have decided to lay out all the previous evidence in order to question the accelerating sea level rise theory.
Other studies
Apart from figures 1 to 4 above the following are also evidence against the accelerating sea level rise theory.
Whilst Figure 3 plotted the “accelerations” with time derived from a single data file Figure 5 plots the end “acceleration” from every data file studied. These started being studied in 2018 and most of the data sets issued by NASA and University of Colorado were used which amounts to over 60 data sets. The equivalent sinusoidal curve is also shown. In figure 5 each point gives the “acceleration” calculated at the end of each analysis. Step changes between consecutive points represent major changes in data.
Figure 5
Figures 6 and 7 are plots of the sea level residuals with best fit quadratic and sinusoidal lines shown. The residuals are the difference between the actual value and the value on a best fit linear line.
Figure 6
Figure 7
The sinusoidal curve is seen to be a much better fit. The reduction in Standard Deviation seems quite small but most of this is due to the large variation caused by the annual and semi-annual signal. The Standard Deviation of the residuals about the y=0 line is 5.83 so the improvement in Standard Deviation is 0.52 for the quadratic curve and 0.70 for the sinusoidal curve.
An Excel Workbook was developed to find the best fitting sinusoidal curve to a set of data. In this an equation of the form in Excel format
Y = CONST+AMP*SIN(((SHIFT+2*X)/PERIOD)*PI())
is used in which Y is the Residual
CONST = a vertical offset that comes about because of the start date for the data (mm) AMP = the amplitude of the sinusoidal curve (mm)
SHIFT = a phase shift (years)
PERIOD = the period of the sinusoidal curve (years)
An initial estimate of the 4 parameters is made and by applying small +/- changes to these parameters and calculating the square root of the sum of the errors squared the process converges on to a best fit curve.
This was applied to the 2007 Start data. The result is shown in figure 8 and is compared to the 1993 finding. There is little change in the parameters of the derived equation. The constant goes from -0.6 mm to -1.1 mm, but this has no effect on the other parameters. The amplitude increases from 3.8mm to 4.2 mm, the phase shift stays the same, and the period reduces from 27.6 years to 27.0 years. Fitting quadratic curves changes the quadratic coefficient from +0.0288 to -0.0243 equivalent to a change in the acceleration of 0.0576mm/year2 to a deceleration of 0.0486 mm/year2.
Figure 8
And finally
What estimates of “acceleration” did the advocates of an accelerating sea level rise make over the last 8 years?
In Ref 1 (2018) a value of 0.084mm/year2 +/- 0.025mm/year2 was quoted based on 1 sigma standard deviation.
In ref 2 (2022) a value of 0.08mm/year2 +/- 0.065mm/year2 was quoted based on 5% and 95% bounds.
In ref 4 (2024) a value of 0.08mm/year2 +/- 0.06mm/year2 was quoted based on 5% and 95% bounds. This paper also stated, “the impact of the 2023 El Niño is minor, demonstrating that the currently measured acceleration is unlikely to vary significantly in the immediate future.”
Note that over 2025 the “acceleration” dropped 0.008 mm/year2 and up to the start of September 2026 the “acceleration” has already dropped another 0.008 mm/year2. In ref 5 I attempted to predict the values of “acceleration” over the next 5 years starting at the end of 2025. Four predictions were made with lower and upper bounds and both with and without a large El Niño. Figure 9 shows the lowest and highest predictions along with the September graph.
Figure 9
The predictions were made on based on the December 2025 data whereas the NASA plot was for the Sept 2026 data. The NASA Data changed in April 2026, and Figure 10 shows the differences between the March 2026 and the April 2026 data with best fit linear and quadratic curves.
Figure 10
Conclusions
All the graphs point to the sea levels varying about the best linear line in a sinusoidal manner with amplitude of around 4mm and a period of around 27 to 28 years.
In the 2022 Nerem et al paper (Ref 2) a graph like figure 3 was included although the date range was 2006 to 2022 (figure 11). Contact was made with one of the co-authors (B.D.H) pointing out that it was an interesting presentation and could it be updated occasionally. No reply was received and as far as can be ascertained no update has been issued.
Whilst rejecting that the sea level rise is accelerating, except for a low value around 0.01mm/year2 commensurate with the long-term tidal gauges, using calculated “acceleration” values using a quadratic fit has proved very useful in following trends and showing that the concept of high accelerations is unsustainable.
Figure 11
Additional Note
After completing this paper, a presentation of the 3 plots with different start dates using a common scale was carried out. This was found to more informative and the results are shown below.
They show the progression of the various peaks and the mirror image between the 1993 and 2007 starts.
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Notes on Data Sources
The various data sets used come from several sources.
NOAA– The individual satellite values were found on
https://www.star.nesdis.noaa.gov/socd/lsa/SeaLevelRise/LSA_SLR_timeseries.php
although this as not been updated since Jan 2025.
Whilst listing the Global data it also gave lists for 24 sub-areas around the world.
NASA– The latest data are found on
In 2018 and for many years the NASA data showed much more information such as number of observations, standard deviation of data and data with and without various adjustments. Now only 2 columns are listed both with the results including the annual and semi-annual signal but with and without the 60-day averaging. Also, NASA have resorted to a 7-day listing instead of a 10-day listing used up to 2025. 10-day listings seem more appropriate as the satellites follow 10-day repeating orbits.
University of Colorado – The data can be found on
They present on average once or twice a year 2 sets of data. These are based on a 10 day spacing and are with and without the annual and semi-annual signal which is useful as without this signal it shows a much smoother plot.
Up to recently the site showed all the historic data and plots but of late only those up to 2018 and the latest are shown.
References
- Climate-change–driven accelerated sea-level rise detected in the altimeter era
R. S. Nerem, B. D. Beckley, J. T. Fasullo, B. D. Hamlingtond, D. Masters, and G. T. Mitchum
https://repository.library.noaa.gov/view/noaa/26954
- Extrapolating Empirical Models of Satellite-Observed Global Mean Sea Level to Estimate Future Sea Level Change
R. S. Nerem, T. Frederikse, B. D. Hamlington
https://wattsupwiththat.com/2026/01/14/measuring-and-analysing-sea-levels-using-
- The rate of global sea level rise doubled during the past three decades
- D. Hamlington, A. Bellas-Manley, J. K. Willis, S. Fournier, N. Vinogradova, R. S. Nerem,
- G. Piecuch, P. R. Thompson & R. Kopp
https://www.nature.com/articles/s43247-024-01761-5
- Technical note on the possible variation in sea level rise accelerations over the next 5-years Alan Welch
- Measuring and Analysing Sea Levels using Satellites during 2023 — Alan Welch
https://wattsupwiththat.com/2024/03/21/measuring-and-analysing-sea-levels-using-satellites-during-2023/
