In addition to seasonal M2 TODS amplitude variations, we are interested in multi-annual trends, such as the ones investigated by Petereit et al. (2019). The analysis of the spectra alone is insufficient to determine these trends. The achieved level of signal peak separation hinders a definite identification of increasing or decreasing signal amplitudes with time. Especially, since, in addition to a higher resolution, it would also require the analysis of the distributions of real and imaginary components around the M2 TODS peak. Unfortunately, the Lomb–Scargle Periodogram does not provide these data.

To analyze trends in the M2 generated signal amplitudes, we divide the preprocessed 10 year time series into sub-series of increasing lengths and apply three different methods to analyze signal amplitudes. Here we rely on methods that have been used in the past to identify TODS ground-based magnetometer observations (Maus and Kuvshinov 2004; Schnepf et al. 2014, 2018). The chosen approach focuses on insights about the dependence of extracted signal amplitudes on the quantity of available data and, as the temporal behavior of the signal is currently unknown, indications for the temporal variation of M2 TODS. Considering the challenge of separating the M2 TODS with its uncertainties from the residual data, it also allows assessing the dependence of the resulting amplitudes and phases on the chosen extraction method.

For our analysis, it is vital that low-frequency signals were removed from the data. If not, the signals may not be separable from the data. Hence, the detrending methods of subtracting the geomagnetic field model and first-order differencing are unsuitable to meet this precondition, especially for the data obtained at Ascension (ASC) (c.f. Fig. 1). For the following part of the study, we rely thus on the residual data obtained after subtracting smoothing splines.

### Phasing and averaging

The first method we applied to identify M2 TODS amplitudes is called phasing and averaging. In this method, each data point is transformed to a new time coordinate related to the phase of the M2 oscillation. The continuous time axis is thereby transformed into a periodic time axis of M2 period length (12h and 25 min). The now overlapping data points are averaged for each minute on the axis. We computed the averages using the arithmetic mean and the median (cf. Fig. 3).

The unique period lengths of the different tidal components create an ever-growing phase shift between the M2 and other tides with each completed oscillation. Assuming that TODS are symmetrical oscillations around the zero value, the signals of tidal constituents other than multiples of the M2 tide are canceled out in the averaging process in long time series. If we further assume that the noise is normally distributed with zero mean, only the mean signal of the M2 tide will remain in the averaging process. We estimate the amplitude from the obtained sinusoidal curve as the halved peak-to-peak difference.

The method is easily implemented and delivers fast results. However, we also want to emphasize the following drawbacks. Due to the made assumptions, it is only applicable if the pre-processing of the observational data can successfully remove all residuals of slowly varying signals. In addition, the smoothness of the obtained sine-shaped average signal depends on the length of the considered time series, the signal-to-noise ratio, and residual signals’ presence and strength. Consequently, signal separation becomes more accurate when analyzing long time series and the influence of seasonal and inter-annual variation decreases. The sinus-shaped average signals thus get smoother with time. Furthermore, the method only provides a rough estimate of the average amplitude but does not account for a possible phase shift of the M2 TODS. This means that a phase shift may mask an increase in the amplitude in long time series and vice versa.

For the analysis of the 10 year time series, we started with time series of 1-year length and increased the size incrementally by 1 year. This was done forward in time, starting with the first year, and backward in time, beginning with the last. The results of this analysis are shown in Fig. 3. When comparing the results obtained using the mean and median, we find a consistent amplitude decrease in both methods for all three observatories. However, for the difference in the magnitude of obtained M2 TODS amplitudes, we find a dependency on the selected station. While the difference between mean and median derived amplitudes is consistently large at CZT throughout the entire time series(\(\approx 0.25~\)nT for all time series length), the difference at ASC and SJG decreases with time (from \(\approx 0.1~\)nT for time series of up to 5 years to almost 0 for more extended time series). When analyzing the time series data (not shown), we find an asymmetric oscillation around the zero-axis in the obtained signal after averaging with the mean, which is not present in the median. Since an offset between median and mean usually indicates a skewed distribution of the given data, we assume that there is either systematic signals in the data which have not been accounted for in the data processing or coastal effects causing distortion or shift of the sinusoidal signal form. One example of such an effect is the presence of ocean currents causing tidal velocities to shift away from the zero baseline, a phenomenon often found in estuaries.

Focusing on the results of the forward analysis, we find that amplitudes decrease with time-series length, which suggests a decrease in all amplitudes over time. This finding is supported by the results obtained from the backward analysis. When comparing the average amplitudes in the first 5-year period (forward analysis) to those of the last one (backward analysis), we find that average amplitudes in the first are indeed larger than those of the last 5-year period. All in all, this validates that there is indeed a perceivable decrease in the amplitudes at the chosen station. However, the large jumps between consecutive years in the first 4 years of the analysis (forward and backward) demonstrate the considerable uncertainty of the method for short time series. Phase shifts, however, have not been analyzed with this method. In principle, this is possible by fitting a sinusoidal model to the obtained averaged sinusoidal curves, but it would add only little value when evaluating the other methods. Then again, the phase can not be neglected, neither as a possible source for the observed TODS temporal behavior nor as a defining feature of this temporal behavior. Especially, since Saynisch-Wagner et al. (2020) have established the relation between changes in the geomagnetic field and ocean conductivity.

### Least squares

The second extraction method is fitting a function *F*(*t*) to the residual time series data employing the least-squares method. The fit functions *F*(*t*) are model the amplitude of TODS over time. Mathematically they are constructed as sums of harmonic functions related to varying tidal constituents, such as the S1, S2, M2. They have the form:

$$\begin{aligned} F(t) = \sum _n A_{Tide} \sin (\omega _{Tide} \cdot t) + B_{Tide} \cos (\omega _{Tide} \cdot t). \end{aligned}$$

(2)

*F*(*t*) models the amplitude of ocean tide induced magnetic field signals at a given time *t* using a sum of harmonic functions with tidal frequencies \(\omega _{Tide}\). Equation (2) uses only radial components of the magnetic field, since TODS are only measurable in vertical direction outside of the ocean (cf. sec. ). The index *Tide* indicates the tidal constituent, such as S\(_1\), \(S_2\) or M\(_2\). The free parameters \(A_n\) and \(B_n\) are determined by the least-squares fit. From these coefficients, we compute the TODS amplitude \(B_{r,n}\) and also the phase \(\phi _n\) of each tidal mode as

$$B_{r,Tide} = \sqrt{A_{Tide}^2 + B^2_{Tide}}$$

(3)

$$\begin{aligned}&\phi _{Tide} = {\left\{ \begin{array}{ll} \arctan \left( \frac{B_n}{A_{Tide}}\right) &{} \text {if } A_{Tide}> 0, \\ \arctan \left( \frac{B_{Tide}}{A_{Tide}}\right) + \pi &{} \text {if } A_{Tide}< 0 \text { and } B_{Tide}>0,\\ \arctan \left( \frac{B_{Tide}}{A_{Tide}}\right) - \pi &{} \text {if } A_{Tide}< 0 \text { and } B_{Tide}<0,\\ \frac{\pi }{2} &{} \text {if } A_{Tide} = 0 \text { and } B_{Tide}>0,\\ -\frac{\pi }{2} &{} \text {if } A_{Tide} = 0 \text { and } B_{Tide}<0,\\ \text {undefined} &{} \text {if } A_{Tide} = 0 \text { and } B_{Tide}=0, \end{array}\right. } \end{aligned}$$

(4)

.

We fitted eight different functions *F*(*t*), or *models*, to the residual time series. Each fit function differs in the number and choice of tidal constituents used to fit the data. The only constant is the fit of the tidal frequency of the M2, the principal lunar tide. Details about the composition of tidal components in each model and exact values of corresponding tidal frequencies used can be found in the Additional file 1. In general, the fit function can be classified into two groups. The first group focuses on fitting the M2 tide, either alone or together with its variations. There, functions include only frequencies corresponding to the M2 tide, its overtides and cyclical amplitude variations, such as semi-annual variation. The simplest example models signal of the M2 tide using two parameters (\(A_{M2}\) and \(B_{M2}\); it is similar to the M2 amplitude determination method for coastal island magnetometer observations of Maus and Kuvshinov (2004). The most complex model in this group, called “M2_overtides_modulation”, includes overtides and their seasonal variation using 240 parameters. Cyclical amplitude modulations are modeled by including sidebands of long periodic modulation frequencies \(f_{mod}\) corresponding to annual and monthly variations. Mathematically, the sidebands are formulated as

$$\begin{aligned} f_{M2,mod} = f_{M2} \pm f_{mod} \end{aligned}$$

The second group of fitting functions is based on the models used by Schnepf et al. (2014) or by the National Oceanic and Atmospheric Association (NOAA). In addition to the frequencies corresponding to the principal lunar tide M2 and its overtides or modulations, further tidal components such as the O1 or S2 are included. While the “Schnepf_2014” model includes 15 tidal components, the “NOAA” model consists of the 37 constituents which usually have the most significant effect on oceanic tidal sea-level signals (Schureman 1958; Parker 2007). Both of these models naturally include low-frequency tides. In theory, these should not be present after the data processing. Hence, the second group of models additionally compares the influence of excluding low-frequency tides on the amplitude of the M2 tide with models containing the suffix “_short.”

The results of the analysis with the simple least-squares method are summarized in Fig. 4. To ease the comparison between the phasing and averaging and the least-squares method, we included the results of phasing and averaging as black lines.

To investigate the robustness of this method, we first focus on the convergence behavior of the presented curves with time-series length. One finding is that the curves of all measured variables, amplitudes and phases alike, converge. Except for the amplitudes identified in the ASC data. The phases are, in general, very robust and all fit functions seem to converge to nearly the same values in few years. For the amplitude curves in CZT and SJG, the fit functions converge after 2 and 4 years, respectively. 4 years also appears to be the time when the difference between the phasing and averaging and the least-squares method becomes negligible. The dependency of shorter time series on the fitting function indicates that signals have not been separated sufficiently.

When focusing on the temporal progression of the curves, we see a change in both amplitudes and phases, a finding that is supported by the comparison of forward and backward analysis (cf. "Phasing and averaging" section). While changes of the average amplitudes are in the order of \({\mathcal {O}}(0.1~\)nT), changes of the phase are in the order of \({\mathcal {O}}(10^\circ )\) and reach values of more than \(100^\circ\) in all three observatory time series for the backward analysis. The temporal variation of the M2 phases is to be expected Saynisch-Wagner et al. (2020). In all time series, the temporal progression of the amplitudes is very similar to the one found with the first method.

Judging from the spectra after the data processing in "Comparison of Data Processing Procedures" section. and the comparison of values obtained after using the mean and the median in "Phasing and Averaging" section , there is no apparent difference in the quality of the three data series. All the more surprising is that the amplitude curves obtained from the ASC data do not converge, especially since the phases do. We find a cluster of curves delivering values similar to method one, but some curves such as the ones corresponding to the label “NOAA” and “M2_overtides _modulation” show a consistently large offset in the forward and the backward analysis. These functions are also quite distinct as the former consists of the sum of the most common partial tides and the latter of a sum of Mx-Tides, i.e., M2 and its overtides, and their temporal variation. In addition, they deliver reliable results for the observations at CZT and SJG. Furthermore, they agree with the general temporal progression of all curves. Therefore, we can only speculate about residual signals in the data interfering with our analysis. A likely origin for these signals is the ionosphere, as we identified signs of ionospheric signals after the detrending methods of first-order differencing and magnetic field model subtraction in ASC (cf. "Comparison of Data Processing Procedures" section).

A major shortcoming of the least-squares method is its sensitivity to the existence of outliers. The implied assumption of the least-squares method is that errors are normally distributed. Thus the likelihood of extreme outliers occurring is very small. The least-squares approach is very efficient in time series with few outliers and fully explained signals. Nevertheless, it is unlikely that a pre-defined fitting function applied to all measuring sites accounts for local differences in TODS, especially when we consider the dissimilarities among the three residual time series, Fig. 1), the challenge of separating signals using short time series and when taking into account that temporal varying amplitudes and nonlinear effects cause further signal peaks.

### Robust least squares

On account of the least-squares method being overly affected by outliers, its general concept was advanced into an iteratively reweighted least-squares (IRLS) (Holland and Welsch 1977; Huber 2004). The IRLS is an iterative optimization approach that reduces the impact of outliers on the overall fit. For details on the setup of the algorithm, we refer to Schnepf et al. (2014), Schnepf et al. (2018) as we followed their lead for the implementation of the third extraction method used in this study.

We used the same eight fitting functions as in the previous section. The approach is computationally very expensive, especially when minimizing fit functions with up to 240 parameters with 10 years of average observations for every minute within the time series. The results obtained with the forward analysis are sufficiently conclusive. Since the backward analysis does not provide additional value, we omitted it consequently for this part of the study. The results are summarized in Fig. 5.

Comparing Figs. 5 and 4, we see the same qualitative temporal behaviour of mean amplitudes and phases. We also find that phase values are in general less sensitive to the chosen fitting function which indicates an increased robustness of phase values over amplitude values. However, apart from these similarities, we find a considerable deviation in the convergence behavior of both amplitudes and phases. While differences between phases were in the order of \({\mathcal {O}}(5^{\circ })\) in all stations when analysing time series longer than four years with the ordinary least squares approach, there is no apparent generalized behaviour when using the IRLS for the analysis. While at CZT the phase values converge almost immediately and deviate by a few degrees, at ASC we have an almost constant spread of \(\approx 20^{\circ }\). At SJG the individual phase curves form a group with a total spread of \(\approx 30^{\circ }\). Also for the amplitudes, we find a larger spread. The spread at ASC stays large with a value of \(\approx 1~\)nT. The spread at CZT converges after \(\approx 4\) years to a group of curves with a spread of \(\approx 0.1~\)nT. For SJG, we find that amplitudes conserve an almost constant spread of \(\approx 0.2~\)nT. In addition, the curves obtained from the SJG data are seemingly separated into two groups for both, amplitudes and phases. This indicates that the algorithm may not be able to identify a global minimum on a flat curve with several local minima.

All in all, we find it plausible that there seems to be a correlation between amplitude M2 signal strength (largest at CZT) and the observed spread in amplitude and phase. Strong signals are generally easier to identify. The “Phasing and averaging” section already indicated a systematic influence of additional signals on the TODS induced by the M2 tide. In view of these results, we find plausible that the amplitude curves obtained from the CZT data converge to a curve with a substantial offset from the curve obtained by averaging the phased data using the mean.

A possible reason for the noticable divergence of the results obtained from the analysis of the ASC data in comparison to the data of the other observatories is the location. In contrast to CZT and SJG, ASC is located in proximity of the geomagnetic equator. Here, the radial magnetic field is much smaller which leads to the fact TODS amplitudes largely defined by the radial geomagnetic field component are reduced to almost zero. With the decreased amplitude, external influences have a larger relative impact which may increase the challenge of a clean signal separation. Hence, for future attempts to analyse TODS originating from coastal island magnetometer data requires designing suitable fit functions for each observatory individually.