From Watts Up With That?
Guest Essay by Dr. Alan Welch FBIS FRAS
Summary. This report presents and analyses the sea level data for 2024. The year 2023 finished with the strong El Niño still causing faster increasing sea level rises but during 2024 the El Niño reduced considerably. The rate of rise is now more stable at about 3.3 mm/year and the so called “acceleration” has resumed its downward trend. The analysis carried out now uses software to carry out Spectral Analysis and to converge on a best fit sinusoidal curve using an iterative approach.
Preamble – Some thoughts regarding “acceleration”
A term that appears many times in sea level reports is acceleration. I feel that unfortunately it has been misused in situations where no clear acceleration exists, but a quadratic curve has been fitted. These curves yield a coefficient for the t2 variable that have the same units as acceleration but need to be applied with caution. Extrapolation is very misused, especially over many decades. For these reasons I have called the resulting acceleration “acceleration” generally throughout the text but have found it useful in using its changing value with time to judge different models.
I must be careful not to create a “pot calling the kettle black” situation as my work involves using sinusoidal curves and prediction. The sinusoidal curves are now backed up by spectral analysis, which has also showed short period cycles due to El Niño effects, longer periods from decadal oscillations and possible very long periods (c1000 years) in some Tidal Gauge data. The prediction item involves showing how the sinusoidal curve would show the variation of “acceleration” with time over a 70-year period up to about 2060 and comparing this with “acceleration” assessments every month or so.
Main Findings.
Data releases via the https://climate.nasa.gov/vital-signs/sea-level/ web site were made during 2024 for January, June, July, August, October and November but although all were analysed this report uses the latest, January 2025, data. The Figure 1 shows the data with linear and quadratic best fits and shows how the term “residuals” is defined.
The residuals (measured values minus linear values) were plotted in Figure 2 together with a quadratic best fit curve and the standard deviation of the errors added. To check this process is carried out correctly the quadratic term is compared with that for the full data and the linear fit checked as being y = 0 x + 0.
Next in Figure 3, the residuals were plotted together with a sinusoidal curve having an amplitude of 4.2 mm and a period of 26 years. The value of 26 years has been used for a couple of years having been originally eyed in.
The standard deviation has improved over the quadratic fit from 3.13 to 3.06. The period of 26 years has been retained so as not to keep changing but a possibly higher value may be more appropriate hence the above figure was repeated but with a period of 29 years and the standard deviation reduces further to 2.99 as shown in Figure 4. Part of the improvement in the standard deviation is also due to the -0.4mm constant in the equation.
The choice of 29 years followed on due to the introduction of two other analytical techniques. Firstly, the fitting of a sinusoidal curve using a convergence process where the four parameters in the equation (constant, amplitude, shift and period) are changed by small steps and an error criterion (square root of the sum of the squares of the differences) reaches a minimum. The process does not guarantee this is the absolute minimum but starting with different parameters may help to confirm this and/or using diagonal small increments by combining two of the parameter increments in pairs. Secondly the use of spectral analysis has been introduced details of which can be found in an Appendix. There are many spectral analysis methods and whilst that used (CLEANest method) may not be the most suitable, checks were performed on a range of simulated curves to judge the suitability of the chosen approach and how to interpret the graphs produced.
Inspection of Figures 3 and 4 shows a tendency for the fitted curve to be similar to a quartic polynomial. Figure 4A shows the result of applying a quartic curve and whilst the SD is now reduced from 3.13 for a quadratic to 3.01 it is still slightly higher than the 29-year period. A danger with polynomials is that near each end of the data the graph can become influenced excessively by the highest order term. Not obvious in this case but extrapolators extrapolate at your peril!!
Figure 5 shows the result of the spectral analysis for the residuals in the NASA analysis. The amplitudes are best judged as relative amplitudes indicating the impact of various periods. (I must thank Tonny Vanmunster of the CBA Belgium Observatory for help with his software. Asking him what the term “theta” used in the program on the y axis is he replied “Related to CLEANest, and the “theta” value shown on the Y Axis: In the CLEANest method described by Foster (1995), the Y-axis of the periodogram represents the power of the signal at each frequency (or period) — but it’s important to note that this is not the same as a classical Fourier power. Instead, the Y-axis in a CLEANest periodogram reflects the amplitude of the cleaned spectral component at each frequency, after the iterative removal of sidelobes caused by uneven time sampling (spectral window effects). You can interpret it as a “debiased” or “cleaned” amplitude, where aliasing has been largely suppressed. So, I would suggest to call it the amplitude or CLEANest amplitude.”
It is the Periods of various peaks that are most useful.
As shown in the appendix when sinusoidal periods involved are nearly close to the period covered by the data the highest point on the spectrum fits quite well but is followed by a long, sometimes slowly descending, tail. Figure 6 homes in on the periods below 12 years where the effects of the El Niños and La Niñas are mainly seen.
Figure 7 shows the El Niño index covering the NASA data range the data being extracted and processed through the Spectral Analysis program producing the spectrum shown in figure 8.
With a peak amplitude between 10 and 11 years what role can sunspots have on this. The migration of sunspots towards the equator over the solar cycle and changes in size combined with the sunspot number produces a daily “Total Solar Irradiance” (TSI) as shown in figure 9 which covers the period of the satellite readings.
The Spectral Analysis of this portion of the TSI resulted in the following.
We need some HUMOUR at this stage. Following the lead of 97% of Climate experts I have fitted the quadratic curve shown above and then extrapolated it to 2100. Get a good supply of suntan lotion in!!
The following graphs, Figure 11 and Figure 13, are probably, in my mind, becoming the most important ones as they progressively grow over time. They plot the “acceleration” values against the date they were determined based on the data from the start of 1993 up to that date. Prior to 2012 the “accelerations” were more chaotic due to the shorter time periods and large swings in data. From 2012 onwards there is a more settled form to the graphs superimposed on which are several “S” shape deviations each occurring because of an El Niño and the effect of which diminishes with time. These occurred in 2015/16 (Very Strong), in 2018/19 and 2023/24 (Strong) and even a weak effect in 2021 is just discernible. Since about 2017 there has been a steady decrease in “acceleration” with short increases over a year at each large El Niño. These combine to result in an average reduction in “acceleration” of about 0.002 mm/year2 per year.
Having stated that I think these graphs are very important I have only found one similar example of this graph (figure 15). This was in a paper by R. S. Nerem, T. Frederikse, B. D. Hamlington (Ref 7) dated March 2022. Unfortunately, the data only goes up to the end of 2020 and uses superseded data. In my paper (Ref 5) I discussed this change in data where some historical data, as far back as 1993, has increased by over 8 mm for some, as far as I know, unexplained reason. Three of the figures are reproduced below to show those changes together with a predicted curve based on a 26-year cycle.
The change of 8 mm is about 2.5 times the annual increase in sea levels. The graph, in Ref 7, was, not unreasonably at the time, used to substantiate the statement that “…the acceleration coefficient becomes stable after 2017” but it would be interesting to see it updated. One thing I have noticed is that whereas a couple of years ago the “accelerations” were quoted in the form 0.083 +/- 0.025 mm/year2, which represented 1σ, they are now quoted in the form 0.09 +/- 0.09 mm/year2 to 0.08 +/- 0.06 mm/year2 representing a 90% confidence interval. These changes now lead to a much lower projected limit. At least this lower limit is in line with my work and the Tidal Gauges.
The quoted slopes, accelerations and comments are from Ref 6, “The rate of global sea level rise doubled during the past three decades”. The note 1 below the values are from ref 6.
“Table 1 Changes in rates and accelerations during the altimeter record
| End Date | Rate (mm/year) | Acceleration (mm/year2) | 2020–2050 Sea Level Change (mm) |
| 2017.99 | 3.3 ± 0.4 | 0.09 ± 0.09 | 171 ± 77 |
| 2018.99 | 3.3 ± 0.4 | 0.09 ± 0.08 | 168 ± 71 |
| 2019.99 | 3.3 ± 0.4 | 0.09 ± 0.08 | 174 ± 64 |
| 2020.99 | 3.3 ± 0.4 | 0.09 ± 0.07 | 169 ± 60 |
| 2021.99 | 3.3 ± 0.3 | 0.08 ± 0.07 | 165 ± 57 |
| 2022.99 | 3.3 ± 0.3 | 0.08 ± 0.06 | 158 ± 55 |
| 2024.00 | 3.3 ± 0.3 | 0.08 ± 0.06 | 169 ± 52 |
- Rate and acceleration estimates for different lengths of the satellite altimeter record. The start year is fixed in 1993 but the end year of the record used varies from 2017.99 to 2024. The last row indicates the rate and acceleration estimates over the current full record. Uncertainty estimates denote the 90% confidence interval. Additionally, the extrapolation of the measured rate and acceleration is used to project the sea level change from 2020–2050.”
If the residuals had followed a sinusoidal variation based on the 29-year sinusoidal curve an equivalent set of “accelerations” could have been determined. Figure 16 shows the “accelerations” as at the end of Jan 2025 with the “accelerations” calculated using quadratic curve fitting to a set of data obtained from a 29-year curve and extended to 2065. Compared with the previous 26-year curve this slightly extends the prediction of when “accelerations” will reach values similar to long range Tidal Gauge data of between 0 and 0.02 mm/yr2. This is shown as occurring at about 2035. The curve as calculated is asymptotic to 0 but it will probably converge on a slightly higher value in line with Tidal Gauges.
Should I pop my clogs before, say, 2030, it would be nice to think someone out there keeps adding new “acceleration” values to Figure 16. My grand kids are only 11, and have not yet acquired the spreadsheet bug, so not quite ready yet to take up the mantle.
Conclusions
The slope of the data has settled down to about 3.3 mm/year.
The “accelerations”, having peaked at just over 0.09 mm/year2 during the period 2017 to 2020, has now reduced to nearly 0.07 mm/year2 in early 2023 at which stage an El Niño caused a short-term blip back up to 0.08 mm/year2 before resuming a downward trend. With time the effect of future El Niños on the “accelerations” would gradually reduce.
The introduction of the spectral analysis and iterative determination of a sinusoidal curve has been a great benefit. Because of this the process has been extended to a much wider range of data, but as yet this has not all been analysed. This amounted to over 50 times the amount of work (spreadsheets and figures) as needed when analysing the NASA data. Therefore, I have spreadsheets coming out of my ears(!) but to fully present it all will need a large amount of extra compiling of results, producing figures, discussing and finalising the conclusions.
The datasets include the following. As and when each dataset is finalised, I intend to issue a report.
NOAA data
The site https://www.star.nesdis.noaa.gov/socd/lsa/SeaLevelRise/LSA_SLR_timeseries.php issues the Sea Level Data every few months. The sets being processed are those with Seasonal Signals removed as measured by the Topex, Jason-1,-2,-3 and Sentinel-6MF Satellites. The data files analysed were from 1993 to approximately the end of September 2024 although a more recent set of data takes the values up to the end of 2024. They cover the global data and the data for 24 sub-areas from the Pacific Ocean down to small seas such as the Adriatic Sea. An interesting presentation is to plot values like “acceleration” or phase shift for each sub-set on an easterly basis. Any trends may help to make judgements on the legitimacy of the derived “accelerations” as being meaningful or mainly a manifestation of the method of determination.
Tidal Gauge data
There are hundreds of Tidal Gauge datasets and those selected include very long sets, such as Brest, those recently discussed on WUWT, such as the NY Barrage and odd ball ones with negative slopes and/or negative “accelerations”. The general approach adopted was similar as that used for the satellite data except that with longer date ranges involved. With this longer (> 100 years) curve the residuals are calculated with respect to this curve. Also, whereas the NASA data used the spectral analysis only for the residuals, for the Tidal Gaude data it will also be applied to the full data to possibly pick up very long-term curves.
North Atlantic, North Sea and Arctic Sea data
These Tidal Gauge datasets were used in Ref 5 to investigate the theory that the Global Sea Levels had an “acceleration” because the satellite coverage was only 95% and the main areas missing include large areas of the North Atlantic, North Sea and Arctic Sea. If these had a decadal oscillation this may show up as a sinusoidal variation. At that time the sinusoidal curve was judged by eye to be about a 26-year period but now a more accurate assessment can be made.
All nine Tidal Gauges will be considered.
Appendix – Spectral Analysis Program
A recent addition to the analysis tools has been the use of spectral analysis software. The program used is called Peranso, described as Light Curve and Period Analysis Software and used mainly in the analysis of variable stars. It was written by Tonny Vanmunster at the Center for Backyard Astrophysics Belgium & Extremadura. As such it has features that are specific to variable stars but can be used to analyse general sets of data.
Due to star magnitudes being such that brighter stars have magnitudes that are more negative the y data for non-astronomical values must have their signs changed. Also, the system refers to Julian Days which may not be relevant and can be changed to Periods in Years.
To use for a general set of x y data an excel file can be set up with x (time) values in the A column, a set of commas in the B column and the variable, with a changed sign, in the C column forming a CSV type file. This data is then copied into a Clipboard.
Nine cases of data input were produced as shown below (Figure A1) to investigate how the system responds and to investigate the suitability for the type of data found in sea level studies.
The top line is for a single oscillation with the longest period, the second line 2 oscillations and the third line 3 oscillations. The first column is all main long-term oscillations, the second column adds an intermediate oscillation and the third column an extra short-term oscillation. The resulting spectral plots are shown below in figure A2.
The “long tail” on the upper row come about from the fact that all the much longer period curves have a very similar RMS error. It requires about 3 oscillations before a more definite peak period is obtained as is shown in row 3.
A further case involved using a very short section of a 1000-year curve. The figures are taken from the Peranso screen and show the input (Figure A3), and spectrum obtained (Figure A4). The peak is shown as at a period of 1111 years and is hardly discernable. This slightly higher period generally occurs when the input is a short portion of a much longer curve. As a peak it is not very obvious, a large range of long period curves would fit nearly as good.
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References (Not all references appear in this report but may appear in follow on reports)
1. https://wattsupwiththat.com/2022/05/14/sea-level-rise-acceleration-an-alternative-hypothesis/
2. https://wattsupwiththat.com/2022/06/28/sea-level-rise-acceleration-an-alternative-hypothesis-part-2/
4. Nerem, R. S., Beckley, B. D., Fasullo, J. T., Hamlington, B. D., Masters, D., & Mitchum, G. T. (2018). Climate-change-driven accelerated sea-level rise detected in the altimeter era. (full text .pdf) Proceedings of the National Academy of Sciences of the United States of America, 115(9). First published February 12, 2018
6. B. D. Hamlington, A. Bellas-Manley, J. K. Willis, S. Fournier, N. Vinogradova, R. S. Nerem, C. G. Piecuch, P. R. Thompson & R. Kopp. The rate of global sea level rise doubled during the past three decades. https://www.nature.com/articles/s43247-024-01761-5
7. R. S. Nerem, T. Frederikse & B. D. Hamlington. Extrapolating Empirical Models of Satellite‐Observed Global Mean Sea Level to Estimate Future Sea Level Change https://repository.library.noaa.gov/view/noaa/54199
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