Abstract
ALL EARTH 2023, VOL. 35, NO. 1, 2176007 https://doi.org/10.1080/27669645.2023.2176007 Epistemic uncertainty in the kinematics of global mean sea-level rise since 1993 and its dire consequences H. Bâki İz Division of Geodetic Science, School of Earth Sciences, The Ohio State University, Columbus, Ohio, USA ABSTRACT ARTICLE HISTORY Received 25 October 2022 Recent studies reported an ambiguous global sea level acceleration during the satellite Accepted 30 January 2023 altimetry (SA) era (1993–2017). New SA data created an opportunity to resolve this issue. In this study, two competing kinematic models to represent global mean sea level anomalies are KEYWORDS compared. The first model consists of an initial velocity and uniform acceleration. The second Satellite altimetry; tide model replaces the initial velocity and acceleration with a trend and a representation of a long gauge; GMSL acceleration; periodic lunar subharmonic of period 55.8 y, which is determined to be statistically significant lunar subharmonics; climate at globally distributed tide gauge records. The models also include parameters for the periodic change effects of lunisolar origin with periods 18.6 y and 11.1 y annual, and biannual variations in 10- day average of globally SA measurements during 1993–2022. Generalized least squares solu- tions yielded updated statistically significant estimates for all the model parameters and their statistics for both models. However, the outcome failed to resolve the ambiguity of uniform acceleration in global mean sea level confounded with the long periodic lunar subharmonic of period 55.8 y during this period. This epistemic uncertainty will have a dire impact on climate change risk assessments as demonstrated through the prospective comparison of both kine- matic models. . . . the real purpose of the scientific method is to make Their limited global distribution is prohibitive in mak- sure nature hasn’t misled you into thinking you know something you actually don’t know. ing a conclusive inference about the GMSL trend and a uniform acceleration. However, TG can serve effec - R.M. Pirsig, The Art of Motorcycle Maintenance. tively as a consilience tool to verify SA findings about the GMSL rise because of their long records. A plethora of investigations during the last two decades indicated 1. Introduction global sea level accelerations and decelerations. Some Evidence for the global mean sea level (GMSL) rising of them include Woodworth (1990), deceleration and th st faster during the 20 and 21 centuries with global acceleration; Douglas (1992) deceleration; Holgate and warming is an important indicator in assessing anthro- Woodworth (2004), acceleration; Church and White pogenic contributions to the climate change (2006), acceleration followed by deceleration; mechanisms. Jevrejeva et al. (2006), acceleration; International In the past, several studies investigated the pre- Panel on Climate Change, (2007), acceleration; th sence of global acceleration during the 20 century Woodworth et al. (2009) acceleration followed by in sea-level rise using Satellite Altimetry (SA) data. For deceleration; Houston and Dean (2011), no accelera- instance, Yi et al. (2015) reported an increase in GMSL, tion; Watson (2011), a regional deceleration. which can be translated into an average acceleration of These spatio-temporal alternating accelerations and 0.28 mm/y since 2010. Davis and Vinogradova (2017) decelerations at TG stations were recognised as early reported an acceleration along the east coast of North as 1990s (Parker, 1992) and are explained effectively by America up to 0.30 mm/y , Dieng et al. (2017) deter- transient/episodic changes due to the inverted barom- mined an increase of about 0.8 mm/y in the GMSL eter, wind, sea surface temperature etc. effects, as well velocity since 2004, equivalent to an average accelera- as long periodic oscillations of the sea level of astro- tion of 0.062 mm/y . More recently, Nerem et al. (2018) nomical origin at multi-decadal frequencies excited by reported 0.084 ± 0.025 mm/y GMSL acceleration since compounding mechanisms of various natural forcings. 1993 inferred from the globally averaged SA time ser- In the past, two different exciting mechanisms for ies. Soon after, Ablain et al. (2019) reported another compounding in relation to climate change were sug- estimate of GMSL acceleration, 0.12 ± 0.07 mm/y . gested. Munk et al. (2002) conjured harmonic beating In contrast to SA, tide gauge (TG) measurements with the necessary conditions for generating subhar- sample only local and regional sea level variations. monics of the periods P and P . The other mechanism A B CONTACT H. Bâki İz h.baki.iz@gmail.com Division of Geodetic Science, School of Earth Sciences, The Ohio State University, Columbus, USA © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 32 H. B. İZ proposed by Keeling and Whorf (1997) involves repeat and Shum (2020) study demonstrated that the recently coincidences and necessary conditions due to the declared uniform acceleration through the analyses of events of near-perfect constructive interference that global SA measurements can be explained equally well occurs at given intervals. by alternative kinematic models based on previously Under both scenarios, ocean and/or meteorological well-established multi-decadal low-frequency sea level forcings materialise as natural broad band sea level variations detected at globally distributed TG stations variations and modulate orbital forcing, such as lunar such as a lunar subharmonic with a period of 55.8 yr. (İz, nodal tide, thus their interplay leading to multi- 2014, 2022), and a 60-year periodicity reported by decadal scale sea level variations in relation to the Chambers et al. (2012). This ambiguity creates an epis- regression of the lunar node, which completes its temic uncertainty in GMSL rise that should be urgently cycle in P = 18.613 y, to produce sub- and super sub- addressed as it will impact climate change risk assess- harmonics as multiples of lunar node period. ments, which will be demonstrated in this study. The compounding of periodic sea level variations Recently, GSFC (2022) made available a new 10-day and their materialisations in sea level variations at average of globally observed SA measurements, which multi-decadal scales is not speculative. For instance, will be referred to as GMSL measurements for brevity. Yndestad et al. (2008) reported that the water-property The new time series includes extended records and time-series show mean variability correlated to their standard deviations as compared to their pre- a subharmonic cycle of the nodal tide of about 74 y. vious version. The availability of the error estimates They also have found significant correlations between for the new SA measurements requires a new statistical dominant Atlantic water temperature cycles and the model for the previously studies kinematic models due 18.6-y lunar nodal tide, and for P/2 = 9.306-year lunar to the demonstrated autoregressive nature of the nodal phase tide. An earlier wavelet analysis by disturbances. Yndestad (2006) identified several lunar nodal sub- The following section describes the GMSL data for and super harmonics in Arctic Sea level, temperature, the period 1993–2022. Subsequently, two competing ice extent, and winter index time series data, including kinematic models are presented with their generalised the signature of nodal harmonics in pole position time least squares, GLS, solutions, followed by an assess- series (Table 1 in Yndestad, Yndestad et al., 2008). More ment of their predicted values. The findings are sum- recently, Chambers et al. (2012) reports the presence of marised in the conclusion section. a quasi-60-y oscillation (close to the 56 y period of the third lunar nodal subharmonic) in GMSL, which could 2. Global mean sea level data also be source of a compounding variable. The presence of the harmonics of the lunar nodal This study utilises GMSL data, shown in Figure 1, tides and the solar radiation variations were exten- which were generated using the Integrated Multi- sively investigated by modelling and estimating the Mission Ocean Altimeter Data for Climate Research amplitudes of the corresponding periodicities in 27 (GSFC, 2022) for the period 1993–2022 consist of globally distributed long TG records by İz (2014). 1076 records (10-day average of global SA measure- Statistically significant signatures of sub and super ments). The updated time series combines Sea Surface harmonics of lunar nodal tides and forced sea level Heights from TOPEX/Poseidon, Jason-1 and OSTM/ variations due to solar radiation are detected in all Jason-2 with all biases and corrections applied and station records. The meta-analysis of the harmonic placed onto a georeferenced orbit. This creates amplitudes from all stations reveals that the effect a consistent data record throughout time, regardless sizes are statistically significant and provide evidence of the instrument used. The GMSL time series is cor- for the harmonic beating/compounding of sea level rected for the effect of Global Isostatic Adjustment changes as a global phenomenon. The compounding (GIA), (Beckley et al., 2016). The new time series, as of the lunar nodal tides and forced sea level changes compared to their previous version, include extended due to solar radiation with other broadband natural records and their standard deviations shown in Error: and forced sea level oscillations is also a plausible Reference source not found. Although the errors are explanation for the recent periodic sea level variations nearly homogeneous, their magnitudes are relevant in SA and TG measurements (İz, 2014). for accurate assessment of the investigated kinematic These effects are consequential in understanding the models and they are essential for the accuracy of the origin of the sea level accelerations and decelerations. İz GMSL to be predicted. Table 1. Model solutions. All uncertainties are 1σ. N/A: Not Applicable. SE: Standard Error, Amp.: Amplitude. Velocity Acceleration SE ARH(1) Solution (mm/yr.) (mm/yr ) Amp. 55.8 yr. (mm) (mm) ρ Model I 3.21 ± 0.04 0.071 ± 0.008 N/A 27 0.7 Model II 3.23 ± 0.03 N/A −6.80 ± 0.77 27 0.7 ALL EARTH 33 −40 −120 −200 1990 1995 2000 2005 2010 2015 2020 2025 Year Figure 1. Globally averaged sea level anomalies as observed by SA (blue) and their ±1σ uncertainties. The series consist of 1076 records during 1993–2022 (10-day averages). 3. Model I: kinematic model to represent the effects whose presence were already established in GMSL anomalies during 1993–2022 with TG series as discussed in the introduction section. a uniform global acceleration For instance, a 60-year period in global sea level variations) was detected in TG records by Chambers A simple kinematic model was proposed and evalu- et al. (2012). Haigh et al. (2011) reported a 59-year ated in recent studies (Ablain et al., 2019; Nerem et al., period generated by the interference of 8.85 y cycle 2018; İz & Shum, 2020). In this study, the model, Model of lunar perigee with 18.6 y nodal cycle at global I, is augmented by representations of annual and semi- scale. İz (2014) also analysed 27 globally distributed annual GMSL variations, long TG records and determined statistically signifi - cant signatures of sub- and super harmonics of h ¼ h þðt t Þv þ ðt t Þ t t 0 t 0 0 0 � � � � �� lunar nodal tides and forced sea level variations 2π 2π þ α sin ðt t Þ þ γ cos ðt t Þ including a long periodic subharmonic with m 0 0 P P m m a period 55.8 years as a global phenomenon. More þ ε t ̇ recently, İz and Shum (2020) demonstrated the pre- (1) sence of the same nodal subharmonic of period 55.8 y period in monthly SA record during 1993– In this representation, h denotes 10-day averaged global SA observations at epochs, t ¼ t . . . t ; where 1 n Because of their similitudes in magnitudes, n is the number of 10-day averages. The intercept h is a uniform acceleration and the 55.8 y nodal subharmo- the reference height of the GMSL defined at the refer- nic are collinear, hence they cannot be estimated ence epoch t ¼ 2005 chosen to conform with the simultaneously. To evaluate the effect of the 55.8 y initial epoch of the previous studies. This designation periodicity at the global scale, the following offshoot is important since the estimated trend of the kinematic of Equation (1) is now considered (Model II) in which models is the initial velocity, v and the trend/velocity the uniform acceleration is replaced with the 55.8 y changes over time in the presence of the uniform periodicity, i.e. acceleration, a. The intercept h is to be estimated � � together with the global initial velocity and uniform 2π acceleration parameters. The model also includes the h ¼ h þðt t Þv þ α sin ðt t Þ t t 0 t 55:8 0 0 0 55:8 � � unknown coefficients of the annual and semi-annual 2π þ γ cos ðt t Þ harmonics denoted by α ^ γ (m ¼ 4). m 55:8 55:8 � � � � �� A competing kinematic model is discussed in the 2π 2π þ α sin ðt t Þ þ γ cos ðt t Þ following section. m 0 m 0 P P m m þ ε (2) 4. Model II: kinematic model to represent the GMSL anomalies during 1993–2022 with The parameters of the Model II were already defined in a 55.8 y periodicity Equation (1) together with their descriptions. Model Sea level variations are multi-causal and occur at I and Model II share the same statistical model dis- various time scales. There are several multidecadal cussed in the following section. Anomaly (mm) 34 H. B. İZ The above structure can be derived by standardising 5. A new statistical model: first-order the observation equations and restructuring the corre- heterogeneously autoregressive disturbances sponding normal equations for a GLS solution. As reported in İz and Shum (2020), the disturbance of the In this model, an estimate of the first-order auto- monthly and globally averaged GMSL disturbances exhi- correlation coefficient ρ is calculated from the residuals bit first-order autoregressive disturbances with an esti- of the OLS and used in the GLS solution in quantifying mated autocorrelation coefficient ρ ^ ¼ 0:9. The origin of the V/C matrix for the SA measurements. The residuals the first-order autocorrelation can be attributed to large- of the GLS solution are then used to calculate a new scale events such as ENSO (Nerem et al., 2018), fusion of ARH(1), which is then used in a GLS solution until there measurements from various SA altimetry missions, and are no changes in the a posteriori variance of the unit smoothing and averaging SA data with 10-day intervals. weight (the standard error of the solution). Iterations In the earlier study by İz and Shum (2020), the following result in an optimal ARH(1) correlation coefficient statistical model has effectively eliminated the impact of between adjacent disturbance terms to be used in the GMSL variations due to the AR(1) in estimating the the final GLS estimation. parameters of the kinematics GMSL anomalies. The cor- In the following sections, ARH(1) statistical model is responding V/C matrix, denoted by Σ, of the first-order used to compare the above kinematics models, Model autocorrelated disturbances, ε ; is given by, I and II. 2 3 2 t 1 1 ρ ρ ��� ρ t 2 6 7 ρ 1 ��� ��� ρ 6 7 Σ ¼ σ � 6 7 (3) . . . . . . . . . . 4 5 6. Concurrent comparison of model I and . . . . . T 1 T 2 T 3 model II ρ ρ ρ ��� 1 Table 1 lists pertinent solution parameters estimated The autocorrelation decreases for increasing time lag using the GLS estimation for the Model I and Model II. because jρj< 1. The correlation between two random The estimated velocity parameters are different, but 2 τ variables ε ^ ε is σ ρ , where τ is the time lag with t t τ the differences are not significant in magnitude. ε ¼ ρε þ u ; 0 � jρj< 1 (4) Because both models use the same data, the null t t 1 t hypotheses for the the differences are not meaningful. The additive noise, u ; is assumed to be uniformly and � Nonetheless, the standard errors of the solutions 2 2 identically distributed (iid), i.e. u iid 0; σ , and σ is u u (a posteriori variance of unit weights) are the same, so the variance of the purely random variations are their iteratively estimated autocorrelation coeffi - (innovations). cients of ARH(1). The estimates for the marginal har- Because the standard errors of the GMSL data are monics shown in Table 2 are also practically the same also provided for the new series (Error: Reference source in both models’ solutions. These outcomes suggest not found) compared to the previous GMSL series, that both models are indistinguishable. a different V/C matrix is required for proper statistical Figure 2 exhibits the time evolution of residuals of representation in this study. The following V/C matrix, both kinematic model solutions, which are included abbreviated as ARH(1), is a first-order autocorrelation here to consolidate the indifferences in the GLS esti- 2 2 structure with heterogenous variances, σ ;��� ; σ . The 1 t mates. Residuals of both model solutions are also correlation between subsequent data is ρ, followed by indistinguishable from each other, as expected, and ρ for two records separated by a third, and so on giving exhibit randomness. Meanwhile, plots of the residuals way to the following first-order heterogeneously auto- against the adjusted anomalies are shown in Figure 3 correlated V/C matrix known as ARH(1), uncover episodic short lasting preponderant trends. 2 3 The correlograms of both models residuals are similar 2 2 t 1 σ σ σ ρ σ σ ρ ��� σ σ ρ 2 1 3 1 T 1 2 t 2 and show that the residual unmodelled anomalies may 6 7 σ σ ρ σ σ σ ρ ��� σ σ ρ 2 1 3 2 T 2 6 7 Σ ¼ 6 7 be autocorrelated in nature other than ARH(1). The . . . . . . . . . . 4 5 . . . . . autocorrelations are strong with lags of up to 2–3 y. t 1 t 2 2 σ σ ρ σ σ ρ ��� ��� σ T 1 T 2 For longer lags, the autocorrelation coefficients are (5) smaller in magnitude, yet they are episodic and Table 2. The amplitudes of the estimated low frequency GMSL variations and their SE using model I and model II. All uncertainties are 1α. N/S: Not Significant (α ¼ 0:05). Nodal Solar Annual Semi-annual Solution Component mm/yr. mm/yr mm mm Model I α −1.60 ± 0.38 N/S −4.12 ± 0.31 N/S γ N/S 2.00 ± 0.34 −1.44 ± 0.30 −1.60 ± 0.03 Model II α −1.69 ± 0.38 NS −4.12 ± 0.31 N/S γ N/S 1.90 ± 0.34 −1.45 ± 0.30 −1.60 ± 0.03 ALL EARTH 35 10 100 −5 −10 −20−16−12 −8 −4 0 4 8 12 16 20 1990 2000 2010 2020 2030 mm Year 10 100 −5 −10 0 1990 2000 2010 2020 2030 −20 −16 −12 −8 −4 0 4 8 12 16 20 Year mm Figure 2. Residual properties of the model I (top) and model II solutions. persistent (autocorrelations in red). The random nature 7. Prospective comparison of model I and of the estimated random noise components, u of the model II autocorrelated residuals, ε , are confirmed through their autocorrelations (blue). The source of unmo- Predicted GMSL rise for the period 2022–2100 were delled short duration autocorrelations that are not carried out using both competing kinematic models. accounted for ARH(1) could be of instrumental origin The underlying algorithm for the calculations was or atmospheric events at global scale and need to be first introduced in sea level studies by İz et al., investigated by the data centres for insight. (2012), and further demonstrated and elaborated Some of the results presented above are redundant in İz (2018). Compared to the traditional least but they are included here to consolidate the outcome squares prediction, which quantifies the expected of the model solutions, which is not conclusive enough values of the predicted anomalies, this approach to validate one model over the other. Given the fact provides the predictions for the actual values to that both models represent the contemporaneous be observed in the future with their prediction GMSL variations equally well, at this point one might intervals (PI) instead of confidence intervals (CI). ask, ‘what difference does it make to identify the correct PIs are therefore better suited in sea level studies model?’. This is the topic of the next section. for risk assessments. 0 0 −10 −10 −20 −20 −60 −40 −20 0 20 40 60 80 −60 −40 −20 0 20 40 60 80 Adjusted Anomaly (mm) Adjusted Anomaly (mm) 0.5 0.5 0 0 −0.5 −0.5 0 5 10 15 20 25 30 35 40 45 50 55 60 0 5 10 15 20 25 30 35 40 45 50 55 60 Lag (yr.) Lag (yr.) Figure 3. Residuals plotted against the adjusted GMSL anomalies to investigate unmodelled effects for model I and model II residuals (top two plots). Their correlelograms are displayed in the second row. Model I Autocorrelation Model I Resid ual (mm) Model II Resid ual (mm) Model I Resid ual (mm) Model II Autocorrelation Model II Resid ual mm) Frequency Frequency 36 H. B. İZ 700 700 600 600 500 500 400 400 300 300 200 200 100 100 0 0 2000 2020 2040 2060 2080 2100 2120 2000 2020 2040 2060 2080 2100 2120 Year Year Figure 4. Predicted GMSL rise for the period 2022–2100 for model I (left) and model II. Prediction intervals are in gray (α ¼ 0:05). Figure 4 displays the predicted results for both mod- GMSL rise and would necessitate interventions in favour els for the period 2022–2100. They are plotted on two of adaptation in decision making at global scale. separate graphs for clarity with the same vertical and horizontal axes for direct comparison. The plots exhibit Notes stark differences between the predicted GMSL anoma- lies, and their uncertainties represented by their PIs 1. Note that, there other competing harmonics of luni- with significance level α = 0.05.Model I predictions solar origin with long periods whose presence were quantified by Iz (2014) at globally distributed TG sta- increases quadratically with larger and larger uncertain- tions with long records. Their statistics, however, do ties because of the uniform acceleration generating not challenge the ambiguity of the uniform GMSL a linearly increasing velocity, whereas Model II predic- acceleration as effectively as 55.8 y period for the tions are stable over time with visually imperceptible globally averaged SA records. variations due to the nodal sub-harmonic with period 2. A detailed plot indicating mission periods is available at NASA’s PODAAC website https://podaac.jpl.nasa.gov/. 55.8 y with a constant trend/velocity. The predicted GMSL anomaly at 2100 by Model I is twice as large, compared to the Model II prediction. Acknowledgments I would like to thank the three reviewers for their precious 8. Conclusion time in reviewing this paper and providing valuable com- ments. This research is partially supported by the Natural Recent analyses of SA time series by Nerem et al. Science Foundation of China (Grant No. 41974040). (2018), Ablain et al. (2019), and others reporting a uniform GMSL acceleration failed to recognise the epistemic uncertainty discussed in detail by İz and Disclosure statement Shum (2020). The current study revisited the topic in No potential conflict of interest was reported by the author. the light of newly available additional five years of global SA data and their standard errors. The outcome of the GLS solutions of two competing models did not Funding find any statistical evidence favouring one model over the other either to alleviate the existing epistemic The work was supported by the Natural Science Foundation uncertainty. Nonetheless, through the comparison of of China [41974040]. the predictions carried out by the two competing kine- matic models, this study demonstrated costly conse- ORCID quences of using a faulty model. On one hand, a faulty Model I for prospective risk H. Bâki İz http://orcid.org/0000-0001-6709-0475 assessments would favour unnecessary interventions (mitigation instead of adaptation). Moreover, if Model References II mirrors reality, then the outcome would introduce further ambiguity, raising the question about the miss- Ablain, M., Meyssignac, B., Zawadzki, L., Jugier, R., Ribes, A., ing impact of the anthropogenic contributions to the Spada, G., Benveniste, J., Cazenave, A., & Picot, N. (2019). GMSL rise. Retrospective analysis of the TG measure- Uncertainty in satellite estimate of global mean sea level ̇ ̇ changes, trend and acceleration. Earth System Science Data, ments together with SA altimetry (İz et al., 2018; İz, 11(3), 1189–1202. https://doi.org/10.5194/essd-11-1189- 2017) may be instrumental to resolve this issue. On the other hand, if the epistemic ambiguity is Bâki Iz, H. B. ,. L., Berry, L., & Koch, M. (2012). Modeling resolved in favour of Model II, then any attempt to regional sea level rise using local tide gauge data. reduce the impact of anthropogenic contributions to Journal of Geodetic Science, Vol. 2(Issue 3), pp. pp. 188– the global climate change would have little effect on 1999. https://doi.org/10.2478/v10156-011-0039-2 Model I Anomaly (mm) Model II Anomaly (mm) ALL EARTH 37 Beckley, B., Zelensky, N. P., Holmes, S. A., Lemoine, F. G., Geodetic Science, 10(1), 29–40. https://doi.org/10.1515/ Ray, R. D., Mitchum, G. T., Desai, S., & Brown, S. T. (2016). jogs-2020-0101 Global mean sea level trend from integrated multi-mission İz, H. B., Shum, C. K., & Kuo, C. Y. (2018). 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All Earth
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Published: Dec 31, 2023
Keywords: Satellite altimetry; tide gauge; GMSL acceleration; lunar subharmonics; climate change