From Watts Up With That?

By Kip Hansen

An Investigation using Tidal Gauge Data

Part 2 – Curve Fitting

By Dr. Alan Welch FBIS FRAS – –  November 2025

Note: This is Part 2 of a series — you might want to read Part 1 first if you haven’t already done so.

Curve Fitting

The individual Tidal Gauges will be processed first.  They cover a wide range of total periods, and some have missing data, generally small gaps but occasionally substantial periods that impinge on the calculations and need to be allowed for.  The Aberdeen working will be presented in full whilst the other Tidal Gauges will be shown in a standard, abbreviated layout.

Aberdeen

The Aberdeen data covers 1862 to 2022 a period of 161 years. The spectral analysis gave 3 curves with a period above11 years (P1, P2 and P3) , namely (84.9, 43.0 and 18.1) years. Associated with these are 3 values of amplitude (amp1,amp2 and amp3) at (32.7, 15.1 and 10.3) respectively. Each of these will have an associated sinusoidal curve given by Equation 1. The 3 values of Amplitude of the curves (AMP1, AMP2 and AMP3) are related to the values (amp1,amp2 and amp3) by Equation 2 which results in the relative values of (a1, a2 and a3) being in the ratios (5.7, 3.9 and 3.2) these being the square root values of (amp1, amp2 and amp3).

The last 3 values (a₁,a₂ and a₃) being relative must be multiplied by a factor, F, to obtain actual Amplitudes of the sinusoidal curves.  F can be guesstimated or if there is a sufficiently number of cycles the extreme range would be close to the range of the individual maximum and minimum amplitudes (A_maxand A_min) of the data resulting in a value of F as shown in Equation 3.

                                     F = (Amax – Amin) / 2 x (a1 + a2 + a3)                                     Equation 3

For the Aberdeen data A_max = 35.7 and A_min = -42.2 giving a range of 77.9

And with               (a₁ + a₂ + a₃) = (5.7 + 3.9 + 3.2) = 12.8

                                       F = 77.9 / 25.6 = 3.04

resulting in Amplitudes (AMP1, AMP2 and AMP3)  of (17.3, 11.9 and 9.7).

These are probably underestimates and the slightly higher values 18, 12.5 and 10 will be tried.

 Figure 14 shows the Aberdeen data (an approximately 8-year moving average) on which the three periods are shown. 

Figure 15 shows the Aberdeen data with the combined values of the 3 sinusoidal curves.  It shows a reasonable fit but will not be pursued any further as the averaged actual data is affected in places by gaps in the initial full data set.  Also, the sinusoidal curve phase shifts have been eyeballed in, and no constant term (CONST) has been included, that is the sinusoidal curves oscillate about the zero residual value.  The shifts (SHIFT1, SHIFT2 and SHIFT3) used are (-30, -20 and -90).

The main deviations appear at approximately Time = 110 (2010).  From 2016 to 2021 there is a lot of missing data as can be seen in the NOAA data shown in the section on Aberdeen above.  Portions of the NOAA data are presented below in Figure 16.

Bergen

The Bergen data covers 1915 to 2022 a period of 108 years with 2 medium gaps in the 1940’s and 1970’s.

                             (P1, P2 and P3)     =      (66.9, 28.9 and 19.2) years

          (amp1,amp2 and amp3)     =     (13.3, 10.2 and 10.1)

                             (a1, a2 and a3)      =     (3.6, 3.2 and 3.2)

                               (A_max and A_min)       =     (32.3 and -40.9)

Applying Equation 3 results in

                                     F = 73.2 / 20.0= 3.66

giving               (AMP1, AMP2 and AMP3)  13.2, 11.7 and 11.7 respectively.

Values of Amplitude 14, 12 and 12 will be tried.

Figure 17 shows the Bergen data (approximately 8-year moving average) on which the three periods are shown. 

Figure 18 shows the Bergen data with the combined values of the 3 sinusoidal curves.  The shifts (SHIFT1, SHIFT2 and SHIFT3) used are (-50, -40 and -90).

The only main deviations appear at approximately Time = 110 (2010) when there is a lack of a peak in the data.  No explanation can be found but Figure 19 is inserted to show the full NOAA data at this time.

Pause for thought

At this stage 2 examples of fitting sets of sinusoidal curves have resulted in a satisfactory result.

Is it a Eureka Moment or am I fooling myself?

The process boils down to using the Spectral Analysis to indicate the main modes considered as being restricted to periods over about 11 years.  Based on these modes a set of sinusoidal curves are formed that use the periods and calculated amplitudes together with the knowledge of the peak minimum and maximum residual values obtained from a roughly 8 year moving average of the residuals.  Phase shifts for each sinusoidal curve are eyeballed in, and a combined curve obtained.  Is this process a bit cyclic or biased so that it produces good looking fits?

The process will be applied to the other 7 shorter period cases and then judgement on the methodology revisited.  The reasons for this are that it may help to judge the suitability of the method and present a fuller picture of 9 Tidal Gauge data sets.

Narvik

With this dataset the remaining records span less than 100 years and analyses may be problematic. Narvik has a large gap in the data early on as well as strange values for the first year or so.  The data file was therefore truncated and the data used starts at 1947.

The partial Narvik data covers 1947 to 2022 a period of 76 years.

                             (P1, P2 and P3)     =      (26.0, 17.6 and 13.4) years

          (amp1,amp2 and amp3)     =     (2.6, 5.0 and 1.9)

                             (a1, a2 and a3)      =     (1.6, 2.2 and 1.4)

                               (A_max and A_min)       =     (28.9 and -30.4)

Applying Equation 3 results in

                                     F = 59.3 / 10.4= 5.7

giving               (AMP1, AMP2 and AMP3)  =    (9.1, 12.6 and 8.0)

Values of Amplitude 9.0, 13.0 and 8.0 could be tried.

Figure 20 shows the Narvik data (approximately 8-year moving average) on which the three periods are shown. 

Figure 21 shows the Narvik data with the combined values of the 3 sinusoidal curves.  The shifts (SHIFT1, SHIFT2 and SHIFT3) used are (-30, -30 and -90).

The result is quite satisfying. 

Reykjavik

The Reykjavik data covers 1956 to 2022 a period of 67 years.

                             (P1, P2 and P3)     =      (37.0, 18.7 and 14.0) years

          (amp1,amp2 and amp3)     =     (12.5, 9.2 and 9.0)

                             (a1, a2 and a3)      =     (3.5, 3.0 and 3.0)

                               (A_max and A_min)       =     (52.8 and -39.4)

Applying Equation 3 results in

                                     F = 92.2 / 19.0= 4.85

giving               (AMP1, AMP2 and AMP3)  =    (17.0, 14.6 and 14.6)

Values of Amplitude 18.0, 15.0 and 15.0 could be tried.

Figure 22 shows the Reykjavik data (approximately 8-year moving average) on which the three periods are shown. 

Figure 23 shows the Reykjavik data with the combined values of the 3 sinusoidal curves.  The shifts (SHIFT1, SHIFT2 and SHIFT3) used are (-80, -16 and -4).

All fits quite well.

Murmansk

The data covers 1952 to 2022 a period of 71 years but has a strange divergence between 1990 and 2010 where there seems to be an increase in sea levels of about 100mm over the general trend (figure 24).  This will affect the spectral analysis, but the process will be completed with the full data.

  (P1, P2 and P3)     =      (65.2, 17.6 and 13.1) years

          (amp1,amp2 and amp3)     =     (143.5, 9.8 and 5.0)

                             (a1, a2 and a3)      =     (12.0, 3.1 and 2.2)

                               (A_max and A_min)       =     (112.5 and -79.7)

Applying Equation 3 results in

                                     F = 192.2 / 34.6= 5.55

giving               (AMP1, AMP2 and AMP3)  =    (66.6, 17.2and 12.2)

Values of Amplitude 18.0, 15.0 and 15.0 could be tried.

Values of amplitude 67, 17.5 and 12.5 could be tried.  These are a bit higher because with a smaller total period there is less chance of the peaks and dips of the 3 curves coinciding.

Figure 25 shows the Murmansk data (approximately 8-year moving average) on which only the longest period is shown.  The 2 short periods are not shown as the strange data between 1990 and 2010 greatly affects the plot. The shifts (SHIFT1, SHIFT2 and SHIFT3) used are (-45, -20 and -100).

Figure 26 shows the Murmansk data with the combined values of the 3 sinusoidal curves. 

The effect of the strange data between 1990 and 2010 shows up with an untypical plot in which the smaller oscillations are dominated by a saw tooth form of the main mode.

Barentsburg

The Barentsburg data covers 1948 to 2022 a period of 75 years.

                             (P1, P2 and P3)     =      (41.6, 17.8 and 13.4) years

          (amp1,amp2 and amp3)     =     (15.2, 16.3 and 14.0)

                             (a1, a2 and a3)      =     (3.9, 4.0 and 3.7)

                               (A_max and A_min)       =     (26.5 and -56.7)

Applying Equation 3 results in

                                     F = 83.2 / 23.2= 3.59

giving               (AMP1, AMP2 and AMP3)  =    (14.0, 14.3 and 13.3)

Values of Amplitude 15, 15 and 14 could be tried.

Figure 27 shows the Barentsburg data (approximately 8-year moving average) on which the three periods are shown. 

Figure 28 shows the Barentsburg data with the combined values of the 3 sinusoidal curves.  The shifts (SHIFT1, SHIFT2 and SHIFT3) used are (-50, -30 and -97).

The two curves in figure 27 generally have the same form.  In this case there were 3 similar peaks which made eyeballing in the phase shifts more difficult.

Lerwick

The Lerwick data covers 1957 to 2022 a period of 66 years with a medium gap at about 2002 and later smaller gaps.

                             (P1 and P2)     =      (21.4 and 12.8) years

          (amp1 and amp2)     =     (5.5 and 3.05)

                             (a1 and a2)=     (2.35 and 1.75)

                               (A_max  and A_min)       =     (17.1 and -20.0)

Applying Equation 3 results in

                                     F = 37.1 / 8.20 = 4.52

giving               (AMP1 and AMP2)=    (10.6 and 7.9)

Values of Amplitude 11 and 8 could be tried.

Figure 29 shows the Lerwick data (approximately 8-year moving average) on which the two periods are shown. 

Figure 30 shows the Lerwick data with the combined values of the 3 sinusoidal curves.  The shifts (SHIFT1 and SHIFT2)) used are (-54 and -25)

The fit is very good.

Tiksi

The Tiksi data covers 1949 to 2010 a period of 62 years.

                             (P1 and P2)     =      (47.4 and 18.1) years

          (amp1 and amp2)     =     (0.835 and 4.28)

                             (a1 and a2)=     (0.91 and 2.07)

                               (A_max  and A_min)       =     (35.6 and -28.6)

Applying Equation 3 results in

                                     F = 64.2 / 5.96 = 10.77

giving               (AMP1 and AMP2)=    (9.8 and 22.3)

Values of Amplitude 10 and 24 could be tried.

Figure 31 shows the Tiksi data (approximately 8-year moving average) on which the two periods are shown. 

Figure 32 shows the Tiksi data with the combined values of the 3 sinusoidal curves.  The shifts (SHIFT1 and SHIFT2)) used are (-54 and -20).

Torshavn

The Torshavn data covers 1957 to 2006 a period of 50 years.

                             (P1, P2 and P3)     =      (38.1, 18.7 and 13.1) years

          (amp1,amp2 and amp3)     =     (1.27, 6.72 and 4.19)

                             (a1, a2 and a3)      =     (1.13, 2.59 and 2.05)

                               (A_max and A_min)       =     (26.5 and -17.7)

Applying Equation 3 results in

                                     F = 44.2/ 11.54 = 3.83

giving               (AMP1, AMP2 and AMP3)  =    (4.33, 9.92 and 7.85)

Values of Amplitude 4.5, 10 and 8 could be tried.

Figure 33 shows the Torshavn data with the combined values of the 3 sinusoidal curves.

The shifts (SHIFT1, SHIFT2 and SHIFT3) used are (-10, -8 and -82).

The fit is not very good due to the short period of the dataset and that there are several breaks in the data.  The plot showing the periods is not shown.    

Before a general discussion it is opportune to apply the methodology developed to the Brest data.

Brest    

The Brest data covers 1807 to 2022 a period of 216 years.  Figure 10 gave the Spectral Analysis peaks for the Residuals.

                             (P1, P2, P3 and P4)     =      (92.8, 36.1, 28.2 and 19.0) years

          (amp1,amp2, amp3 and amp4)     =     (16.6, 8.8, 15.6 and 9.0)

                             (a1, a2, a3 and a4)      =     (4.1, 3.0, 3.9 and 3.0)

                               (A_max and A_min)       =     (50.0 and -38.4)

Applying Equation 3 results in

                                     F = 88.4 / 28.0= 3.16

giving               (AMP1, AMP2, AMP3 and AMP4)  =    (12.9, 9.5, 12.3 and 9.5)

Values of Amplitude 14.0, 10.0 13.0 and 10.0 could be tried.

Figure 34 shows the Brest data (approximately 8-year moving average) on which the four periods are shown. 

The shifts (SHIFT1, SHIFT2, SHIFT3 and SHIFT4) used are (-55, -70, 101 and -120).

All fits quite well except at the times of missing data, i.e. around 1850 and1950.

Discussion

The main finding is that every Tidal Gauge exhibits 2 or more oscillations in the residual sea levels between 11 and 100 years in addition to the many below 11 years that occur due to irradiance variation or El Niño/La Niña cycles.  Above a period of 20 years these oscillations vary from gauge to gauge in periods and the combined effect is difficult to ascertain but not measuring them over the northly seas would impinge on the satellite “global” readings.

A query still exists into the validity of the methodology used. 

A plot is shown in figure 35 that displays all the Tidal Gauge combined sinusoidal curves from 1993 to 2025 except for Murmansk due to its possible bad data and Brest as this was an extra study.  There is a similarity between the curves.

The 8 curves are averaged and plotted in Figure 36 at the same scale.  It shows an approximate  18-year cycle.

The significance of the 18-year cycle can be appreciated by plotting the Amplitudes and Periods of each set of sinusoidal curves as shown in figure 23 (plus Murmansk).  There are 2 clusters.  One at a Period of 13.5 years with 6 of the 9 curves involved and one at 18 years with 8 of the 9  curves involved with the 9^th value being 21.4 years.   The second cluster matches the 18-year cycle in figure 36.

The commonality of periods may be an indication of one or more shortish decadal oscillations.  Values in the range of about 25 to 30 years were being searched for based on the satellite analyses.  The period of satellite coverage was only 32 years, and the area not covered, whilst being mainly the upper Atlantic regions, also covered other parts of the Arctic Sea and smaller sea areas surrounding the Antarctic region.  It is not unreasonable that the areas unmonitored could induce sinusoidal variations in the “global” results.

The instigation for this work was the sinusoidal variation in the range of 26 to 29 mm/year² found whilst studying NASA Satellite Data.  This oscillation  was determined for the residual values calculated from the linear best fit of the data.  The Tidal Gauge results are for the residuals calculated from a quadratic curve over, in most cases, the entire data set.  Over the short (33 years) period of the satellite data the deviation of the best fit from a straight line was assumed to be minimal but as the satellite period increases it may become more important to consider it.

The latest NASA data was first looked at.  Unfortunately, there has been a hiatus, due to political reasons, and before that a reduction in the amount of processed data issued.  Therefore, the April 2025 data will be used as that is the last set with a fuller form of data I have analysed and reported on since starting in 2018.  Figure 38 shows the April 2025 data on which best fit linear and quadratic curves have been fitted.

Fitting quadratic and sinusoidal curves to the residuals (actual values – straight line values) resulted in figures 39 and 40.  Note statistically the sinusoidal is more accurate than the quadratic when judged on the errors.

Four spectral analyses were performed

• Full data (As shown on Figure 38 – blue line)
• Residuals (As shown on figures 39 or 40 – blue dots)
• Differences of Residuals to Quadratic Curve (Figure 41)
• Differences of Residuals to Sinusoidal Curve(Figure 42)

Results for the 4 Spectral Analyses are shown in figures 43 to 46.

What do figures 43 to 46 say?  All 4 figures have several peaks below approximately a 10-year period, but it is the rest that are of interest in this study.

On figure 43 there are 3 peaks indicated.  First a very long-term peak for which the actual period cannot be determined due to the short span of the readings.  Then a 16.9-year period in line with the periods found with all the Tidal Gauges.  And lastly a 11-year period possibly due to irradiance variation (solar cycle).

Figure 44 points to the 29-year cycle that is being investigated and again a peak around 10 years.  As stated above this is obtained when working relative to a straight line and arose when making judgement on the use of quadratic curve fitting and the associated extrapolation.  Sinusoidal curve fitting was studied as an alternative with all this being reported in a number of papers, the last being https://wattsupwiththat.com/2025/06/06/measuring-and-analysing-sea-levels-using-satellites-during-2024/ that covers up to the end of 2024.

Figure 45 shows possibly 2 combined closely positioned peaks around 14 to 15 years.

Figure 46 has a single, very sharp, peak at 10.9 years  again possibly due to irradiance variation.  Figures 41 and 42 at first glance look very similar but the sinusoidal curve, being a slightly better fit, produces a clearer spectrum.

The overall conclusion is that the 29-year variation is to do with the work involving the linear fitting but there is a general variation of about 18 years in all the tidal graphs which shows up in the full satellite data.

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