Nonlinear absolute sea-level patterns in the long-term-trend tide gauges of the West Coast of North America

Abstract The research issue of which are the present relative and absolute rates of rise and accelerations for North America is here addressed. The data of the 20 long-term-trend (LTT) tide stations of the West Coast of North America with more than 80 years of recorded data are shown. The absolute rates of rise are computed by considering the absolute vertical velocity of Global Navigation Satellite System (GNSS) antennas near the tide gauges, and the relative rate of sea-level rise from the tide gauge signals. The 20 LTT stations along the West Coast of North America show an average relative rate of rise of -0.38 mm/yr., an average acceleration of +0.0012 mm/yr2, and an average absolute rate of rise of +0.73 mm/yr. This is the first paper publishing a comprehensive survey of the absolute sea-level rates of rise along the West Coast of North America using the reliable information of relative sea-level rates of rise from LTT tide gauges plus the absolute subsidence rates from different GNSS antennas close to the tide gauge installations.


Introduction
Because sea levels oscillate with well-known periodicities in the 60-year range, like other climate parameters [1,2], more than 60 years of continuous recording from the same tide gauge, without any major perturbation, are needed to compute a reliable slope by linear fitting, and more than 90 years are needed to compute a reliable acceleration by parabolic fitting. There are 20 Long Term Trend (LTT) tide stations along the West Coast of North America, from Alaska to Panama. These stations are Ketchikan, The coupling of relative sea-level data from tide gauges and data of the absolute position of antennas from satellite permits to attribute the relative sea-level rise of a specific coastal site to the growth of the water volume or the sinking of the land [7].
The Global Positioning System (GPS) time series, from a constellation of satellites which is used for navigation and measurements of precise geodetic position of antennas, are given and analysed by different providers, such as SONEL, [6], Nevada Geodetic Lab (NGL) [8], or [9], and National Aeronautics and Space Administration (NASA) Jet Propulsion Laboratory (JPL) [10]. While the analysis of sea level data is straightforward, the analysis of GNSS data is more complicated hence there is a need to use multiple providers.
Discontinuities, outliers, seasonality, skewness, and heteroscedasticity are common problems in the estimation of velocities from the GNSS coordinate time series [9]. Additionally, subsidence patterns may also be genuinely non-linear also over the short time windows typically covered by the GNSS time series, as it is the case in areas subjected to abrupt crustal movements such as earthquakes [11].
Except in the very few cases where the GNSS antenna is co-located with the tide gauge, and precise leveling is ensured between the GNSS antenna and the tide gauge instrument, there is no guarantee that the absolute vertical velocity of an inland GNSS antenna is an accurate estima-tion of the absolute vertical velocity of the tide gauge instrument. This aspect has been addressed by [12].
The GNSS monitoring of antennas is a much better estimation than a global glacial isostatic adjustment (GIA) model computation. It is indeed quite popular to correct the relative sea-level rise trend by a GIA computation such as [13,14]. However, global GIA models account for only one of the many components of land motion, completely neglecting any possible crustal movement. The GIA correction has been seriously questioned by [15].
It is well accepted that the correction of the relative rate of rise of the sea level by the absolute velocity of a GNSS antenna nearby the tide gauge returns the absolute rate of rise of the sea levels with higher accuracy [16]. If the GNSS correction is more accurate than the correction by a global GIA model such as [13,14] that does not include any regional subsidence or uplift, nevertheless technicalities limit the accuracy of the GNSS vertical velocities in a specific location, and nearby antennas may also exhibit a strongly variable pattern of subsidence. Hence, in the following sections, one relative MSL result will be coupled to multiple GNSS results, for every tide gauge location.
The difference between the GNSS computations by different providers is the different method of data analysis, the different realignments, and breakpoints, and (in the case of antennas still active) also the number of data points considered. NGL is usually more up-to-date in their analysis, which is based on a much larger number of antennas.
The total number of GNSS antennas considered by SONEL for the entire world is 493 (June 6, 2018). JPL has a few more stations. The total number of GNSS antennas considered by JPL for the entire world is 2822 (June 6, 2018). NGL has many more stations than JPL. The total number of GNSS antennas considered by NGL for the entire world is 15277 (June 6, 2018).
Although in principle the computation of the subsidence rate from a GNSS position time series should be straightforward, technicalities such as satellite drift, break-points alignments, resolution of discontinuities, make the evaluation of the subsidence rate subjected to a much larger uncertainty than what is shown by the different providers, that only compute the fitting accuracy.
While nearby GNSS antennas may, in some locations, suffer from dramatically different subsidence rates, the same GNSS antenna should have about the same subsidence rate from different providers. This is not the case.
While it is impossible to know the real accuracy of every estimation, it may only be judged the consistency in between the subsidence rates provided by the different providers SONEL, JPL and NGL as a measure of reliability. Additionally, more weight may be given to the GNSS antennas closer to, rather than farther from a tide gauge location.
As the NGL network is much more widespread and the data set is generally more complete, usually, the subsidence rate of the tide gauge for every location is taken as the NGL value of a co-located GNSS antenna, or, if no GNSS antenna is close enough, an average of the NGL values of nearby GNSS antennas.
While the chosen value for the subsidence rate at a tide gauge is still somehow arbitrary, this is the best opportunity presently available. Even if the inaccuracy is sometimes larger than the trend, the proposed absolute subsidence rate and absolute rates of rise of the sea levels are the best possible option available.
If GNSS data is lacking, the Glacial Isostatic Adjustment (GIA) vertical velocities VM2 from [13,14] are used. This computation does not include the contribution of local subsidence, and other regional crustal movements, that may be large.
Even though climate models predict accelerated sea level rise, it is well established that long-term-trend tide gauge measurements show that there has been no detectable acceleration in the rate of sea level rise , just to name a few.

Method
Two regressions are usually applied to the measured relative sea levels of a tide gauge record to compute the relative sea-level rate of rise and acceleration.
The rate of rise of the sea level is historically computed as the first order coefficient of linear regression. A linear regression: returns the sea level rate of rise u as slope B. The acceleration of the sea level is computed since more recently as twice the second-order coefficient of linear regression. A quadratic regression returns the acceleration a taken as 2·C. Figure 1 presents an example of the MSL of Brest. In Figure 1.a and b are the raw data with applied linear and parabolic fittings. In a the gaps (264 missing months) are filled, in b the gaps are not filled. The R 2 is also given as an estimation of the fitting accuracy. This is, unfortunately, only an estimation of the fitting uncertainties. The major issue in computing the rates of rise and accelerations of the sea level is the lack of good quality data, with often records that are too short, with gaps, and originating from different tide gauges of different land and sea contributions, sometimes also misaligned each other. With gaps filled, the rate of rise and the acceleration are 1.0981 mm/yr. and 0.0095 mm/yr 2 . With the gaps, the rate of rise and the acceleration are 0.9984 mm/yr. and 0.01264 mm/yr 2 .
Some providers of sea level analyses, such as [4], use slightly different approaches, where the raw data is first processed and cleared, and then two fittings having a common first-order coefficient are applied, and a different x is used. Obviously, the computed rates of rise and accelerations do not practically change, as MS Office Excel provides the same rates of rise and accelerations of other statistical methods applied to the same data.
For the case of Brest, from [4], a linear regression in x = (date -1913.62), i.e., 1913.62=1913/8, returns y = B + M·x y = 6997.783 + 0.997·x mm while a quadratic regression in x returns y = B' + M·x + A·x 2 y = 6973.966 + 0.997·x + 0.00635·x 2 mm [4] then prefers to use a 95% confidence interval rather than R 2 as a measure of the still only statistical fitting accuracy. If the record does not satisfy quality and length requirements, the computation of the rate of rise and the acceleration is wrong whatever may be the confidence interval.
The slope is thus 0.997 mm/yr. vs. the 0.9984 mm/yr. of the simple linear fitting of the raw data with gaps, for a difference of 0.001 mm/yr., while the acceleration is thus 0.01269 mm/yr 2 vs. the 0.01264 mm yr 2 of the simple parabolic fitting of the raw data with gaps, for a difference of + 0.00005 mm/yr 2 .
By taking as the present (2018) sea level rate of rise and acceleration 0.9984 mm/yr. and 0.01264 mm/yr 2 , the constant acceleration sea level rise by 2100 is 124 mm. By taking as the present (2018) sea level rate of rise and acceleration 0.997 mm/yr. and 0.01269 mm/yr 2 , the constant acceleration sea level rise by 2100 is also 124 mm. Considering the present debate is about the missing meter, or meters, of sea-level rise by 2100, and no-quality short segmented records are given same relevance of high-quality records originated by a single tide gauge instrument that has recorded continuously in the same location over more than a century, these differences may be considered irrelevant.
For better details, if needed, of the data provided in the next sections, that originates from www.sealavel.info, the reader is referred to the web site, and to the work by [46].
The subsidence rate of the land is historically computed as the first order coefficient of linear regression. The linear regression is also applied to the absolute vertical position of the GNSS record for antennas located nearby tide gauge installations. The linear regression now returns the absolute velocity w as the slope B. In this case, the major source of uncertainties are the many corrections of drifts and misalignments, additional to the short records. Better information is here given in the NGL web site.
The absolute rates of rise of the sea levels are then computed as v=u+w [16].

Results
Here

Ketchikan, AK, USA
Ketchikan is the south eastern most city in Alaska. The MSL trend at Ketchikan, AK, USA, Figure 3, is -0.33 mm/yr. with a 95% confidence interval of ±0.22 mm/yr., based on MSL data from 1919/1 to 2017/12. The acceleration is -0.01808±0.01746 mm/yr 2 . The closest GNSS Stations from SONEL are AIS1, with absolute vertical velocity 0.88±0.33 mm/yr.,AIS5 with a signal not robust, AIS6, with no data. AIS6 has a distance-totide-gauge of 29,331 m, AIS1 has a distance-to-tide-gauge of 29,308 m, and AIS5 has a distance-to-tide-gauge of 29,308 m. Hence, all these GNSS antennas are relatively far from the tide gauge.
The maximum and minimum subsidence rates are 0.819 and -1.772 mm/yr. respectively.
The closest GNSS Stations from SONEL are BIS1, with absolute vertical velocity 1.80±0.51 mm/yr., BIS5 of signal not robust, BIS6, with no data. BIS6 has a distance-totide-gauge of 24,948 m, BIS1 has distance-to-tide-gauge of 24,922 m, BIS5 has distance-to-tide-gauge of 24,922 m. Hence, also here the GNSS antennas are relatively far from the tide gauge. According to JPL, BIS1 has absolute vertical velocity 1. From the NGL results a likely absolute vertical velocity of the tide gauge instrument is taken as the average subsidence rate of the nearby GNSS antennas, +1.498 mm/yr. The maximum and minimum subsidence rates are 2.280 and 0.394 mm/yr. respectively.

Juneau, AK, USA
The MSL trend at Juneau, AK, USA, Figure 5, is -13.16 mm/yr. with a 95% confidence interval of ±0. 35   The maximum and minimum subsidence rates are 17.745 and 14.820 mm/yr., respectively.
The closest GNSS Station from SONEL is AV09 has an absolute vertical velocity of 3.50±0.30 mm/yr.
The distance-to-tide-gauge is 950 m. According to JPL, AV09 has an absolute vertical velocity of 2.974±0.264 mm/yr. According to NGL, AV09 (data 2004.3450 to 2018.4367) has an absolute vertical velocity of 3.779±1.247 mm/yr.
A likely absolute vertical velocity for the tide gauge instrument is taken as the NGL value for AV09, of 3.779 mm/yr.

Prince Rupert, Canada
The MSL trend at Prince Rupert, Canada, Figure 7, is +1.17 mm/yr. with a 95% confidence interval of ±0.23 mm/yr., based on MSL data from 1909/1 to 2016/12. The acceleration is 0.01484±0.01463 mm/yr 2 .  As the NGL value for BCPT suffers from large uncertainty, because of the short record, a likely absolute vertical velocity for the tide gauge instrument is taken as the NGL value for BCPR, of 1.369 mm/yr.

Point Atkinson, Canada
The

Vancouver, Canada
The MSL trend in Vancouver, Canada, Figure 9, is +0.49 mm/yr. with a 95% confidence interval of ±0.22 mm/yr., based on MSL data from 1909/11 to 2016/12. The acceleration is 0.0251±0.0141 mm/yr 2 . There is no nearby GNSS dome by SONEL. As a rough estimation of the absolute vertical velocity of the tide gauge, the JPL result for CHWK and CWAK, southeast of Vancouver, Canada, and P440, south of Vancouver, Canada, or also the NGL results for CHWK and P440 are considered.
Same as Point Atkinson, Canada, from the average of the NGL results, a likely absolute vertical velocity for the tide gauge instrument is taken as 0.556 mm/yr. The maximum and minimum subsidence rates are 1.162 and -0.050 mm/yr. respectively.
The distance-to-tide-gauge is 9,431 m. According to JPL, ALBH has absolute vertical velocity 0.17±0.215 mm/yr. According to NGL, ALBH (data 1996.0000 to 2018.4367) has absolute vertical velocity 0.818±0.472 mm/yr. From the NGL result for ALBH, the absolute vertical velocity of the tide gauge is taken as +0.818 mm/yr.
The closest GNSS Stations from SONEL are TFNO has an absolute vertical velocity of 1.19±0.47 mm/yr. UCLU has an absolute vertical velocity of 2.99±0.19 mm/yr. TFNO has a distance-to-tide-gauge of 343 m, UCLU has a distance-to-tide-gauge of 37,070 m. From the NGL results for TFNO, which is almost colocated with the tide gauge, the absolute vertical velocity of the tide gauge is taken as +0.716 mm/yr.
The nearby GNSS Station from SONEL is SC02 has absolute vertical velocity 0.23±0.24 mm/yr.
From the NGL results for SC02, the absolute vertical velocity of the tide gauge is taken as +0.192 mm/yr.
The closest GNSS Stations from SONEL are: SMAI, with no data, SEAT of absolute vertical velocity -0.99±0.22 mm/yr., SSHO with no data.
SMAI has distance-to-tide-gauge of 8,690 m, SEAT has distance-to-tide-gauge of 5,900 m, has SSHO distance-totide-gauge of 10,520 m.

Neah Bay, WA, USA
The MSL trend at Neah Bay, WA, USA, Figure 14, is -1.69 mm/yr. with a 95% confidence interval of

Astoria, OR, USA
The MSL trend at Astoria, OR, USA, Figure 15, is -0.14 mm/yr. with a 95% confidence interval of ±0.33 mm/yr., based on MSL data from 1925/2 to 2017/12. The acceleration is 0.01480±0.02760 mm/yr 2 .  From the NGL results for TPW2, which is co-located with the tide gauge, the absolute vertical velocity of the tide gauge is taken as +0.340 mm/yr.

Crescent City, CA, USA
The MSL trend at Crescent City, CA, USA, Figure 16, is -0.78 mm/yr. with a 95% confidence interval of ±0.30 mm/yr., based on MSL data from 1933/1 to 2017/12. The acceleration is -0.00857±0.02672 mm/yr 2 . CACC has a distance-to-tide-gauge of 2 m, while PTSG has a distance-to-tide-gauge of 7,195 m.
According to JPL PTSG has an absolute vertical velocity of 1.88±0.363 mm/yr. From the NGL result for CACC, which is co-located with the tide gauge, the absolute vertical velocity of the tide gauge is taken as +2.790 mm/yr.

San Francisco, CA, USA
The MSL trend at San Francisco, CA, USA, Figure 17, is +1.47 mm/yr. with a 95% confidence interval of ±0.13 mm/yr., based on MSL data from 1854/7 to 2017/12. The acceleration is 0.01406±0.00619 mm/yr 2 . SBRB has a distance-to-tide-gauge of 14,210 m, UCSF has a distance-to-tide-gauge of 4,850 m, PBL1 has distance-to-tide-gauge of 6,500 m, TIBB has distanceto-tide-gauge of 9,551 m and SBRN has Distance to Tide 14,235 m.
The distance-to-tide-gauge is 8,615 m. However, much closer is the GNSS antenna of CASM, distance-to-tide-gauge of 2,816 m.
From the NGL results for CASM, that is much closer to the tide gauge even if not exactly co-located, the absolute vertical velocity of the tide gauge is taken as -0.927 mm/yr. This result makes the Santa Monica result consistent with the nearby tide gauge patterns and subsidence data.

Los Angeles, CA, USA
The MSL trend at Los Angeles, CA, USA, Figure 19, is +0.99 mm/yr. with a 95% confidence interval of ±0.24 mm/yr., based on MSL data from 1923/12 to 2017/12. The acceleration is 0.01773±0.01924 mm/yr 2 . The closest GNSS Stations from SONEL are CRHS of no data, TORP of no data, VTIS of absolute vertical velocity -0.15±0.14 mm/yr. CRHS is far from the tide gauge, at a distance-to-tidegauge of 11,430 m. VTIS is closer, at distance-to-tide-gauge of 2,168 m. TORP is mid-way at distance-to-tide-gauge of 10,230 m. According

Balboa, Panama
The MSL trend at Balboa, Panama, Figure 22, is +1.44 mm/yr. with a 95% confidence interval of ±0.21 mm/yr., based on MSL data from 1908/1 to 2017/12. The acceleration is -0.00548±0.01504 mm/yr 2 . PMPA is closer to the tide gauge. From all the NGL results, the absolute vertical velocity of the tide gauge is taken as -3.400 mm/yr. maximum and minimum are 0.908 and -7.708 mm/yr., respectively. Table 1 and Figure 23 present a summary of the sea level and GNSS results for the LTT stations of West North America. u is the relative sea-level rise, w is the absolute vertical velocity at the GNSS antenna nearby the tide gauge, and u=v+w is the absolute sea-level rise. The table also proposes as w* the Glacial Isostatic Adjustment (GIA) vertical velocities VM2 from [13,14].

Discussion
The GIA correction does not appear reasonable. Glacial isostatic models are of little help here to understand the absolute sea level rises, suggesting on average an uplift velocity of -0.44 mm/yr. while the vertical velocity from GNSS is +1.01 mm/yr. Especially areas of high uplift, or regional subsidence, suffer from very poor accuracy in estimating the vertical velocity [15]. The average relative rate of rise is -0.38 mm/yr., the average acceleration is +0.00120 mm/yr., and the average absolute rate of rise is +0.73 mm/yr. The acceleration result is consistent with other global and regional estimations from LTT stations such as [17,18] or [19,20]. [19] and [20] recently reported as the latest average acceleration of worldwide data sets is still very close to zero. The Mitrovica's 23 gold standard tide stations with minimal vertical land motion have average acceleration +0.0020±0.0173 mm/yr 2 . The Holgate's nine ex-cellent tide gauge records of sea-level measurements have average acceleration +0.0029±0.0118 mm/yr 2 . The NOAA's 42 U.S. long term trend tide stations of 2011 have average acceleration +0.0025±0.0308 mm/yr 2 . The California-8 long term trend tide stations have average acceleration +0.0014±0.0266 mm/yr 2 . The LTT stations of the West Coast of North America have acceleration values on average positive, but of the order of the nanometers per year squared, similarly to the other data sets.

Conclusions
The GNSS monitoring of the position of antennas is superior to the GIA model computations to assess vertical land velocities. The GIA is a global isostatic factor of theoretical model dimensions. The GNSS values are records of actual site-specific crustal movements. However, the technique still suffers from major uncertainties (differences between the estimates by different providers often larger than the trend). Although the GNSS measurements of vertical velocity still leave much to be desired, they are better than model estimates, based on GIA, of the rate of rise of the land. A particularly valuable part of the paper is the display of raw data on the relative changes of sea levels measured by tide gauges. A crisp summary of the results of linear and quadratic fits to the data is tabled. The measurements at the tide gauges are the best way to understand sea-level changes. These measurements show a stable pattern of mild rising sea levels with negligible accelerations mostly explained by the sinking of the tide gauge instrument, in the 20 LTT tide gauges of the West Coast of North America.
As the GNSS values chosen are far are still uncertain and linked with very great scatter values, the absolute rate of rise result is less reliable than the relative rate of rise and acceleration results.
As the absolute rate of rise of the sea levels from thermal expansion of ocean waters and mass addition from melting of ice on land is small, in uplifting areas, sea level is decreasing, and the contrary happens in subsiding areas. The influence of earthquakes, same as every other disturbance in the tide gauge signals, such as change of position of the tide gauge, or infrastructure extension changing the sea level pattern, should be taken into consideration.