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

The paper provides an estimate of the latest relative and absolute rates of rise and accelerations of the sea levels for the East Coast of North America. The computation is based on the long-term trend (LTT) tide gauge records of the relative sea levels and the Global Navigation Satellite System (GNSS) time series of the absolute position of xed dome nearby the tide gauges. The GNSS result is used to infer the subsidence or uplift of the tide gauge instrument. The data of 33 LTT tide stations with more than 80 years of data are shown. The average relative sea-level rise is +2.22 mm/yr. subjected to a small, positive average acceleration of +0.0027 mm/yr2. The average absolute velocity of the tide gauge instruments is -0.52 mm/yr. translating in an average absolute sea-level rise of +1.70 mm/yr. This is the rst paper publishing a comprehensive survey of the absolute sea-level rates of rise along the East Coast of North America using the reliable information of relative sea-level rates of rise from LTT tide gauges, plus the absolute subsidence rates fromGNSS antennas that are close to the tide gauges installations.


Introduction
The absolute sea-level rise is computed by correcting the relative sea-level rise measured by a tide gauge instrument by the absolute vertical motion of the instrument, either modeled [1] or measured [2]. Système d'Observation du Niveau des Eaux Littorales (SONEL) [3], National Aeronautics and Space Administration (NASA) Jet Propulsion Laboratory (JPL) [4], and Nevada Geodetic Lab (NGL), [5], also [6], provide estimations of the absolute (geocen-  [7]. While an analysis of the PSMSL data permits to estimate the relative rate of rise of the sea levels for these stations, the analysis of the SONEL data permits to estimate the absolute vertical velocity nearby some of the stations, and compute for these stations an absolute rate of rise.
There are requirements for determining the trend, in case of tide gauges length of the period, completeness, quality, and additionally, in case of the GNSS antennas, the stability of the solution. In case of tide gauges, the sea levels oscillate with well-known periodicities in the 60year range, like other climate parameters [8,9], 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 tting, and more than 90 years are needed to compute a reliable acceleration by the parabolic tting. SONEL computes the absolute rates of rise of the sea levels from their GNSS data and the tide gauge data from PSMSL. Only 83 locations are shown worldwide in their latest analysis if a maximum time window 1900 to 2013 is selected. Only 9 locations are shown along the East Coast of the US and Canada, only Halifax in Canada, then 9 locations in the US, Newport, New York, Philadelphia, Atlantic City, Annapolis, Baltimore, Washington DC, Charleston, Key West.
As there are 33 Long Term Trend (LTT) tide stations along the East Coast of North America where their measured MSL relative to the tide gauge instrument are given by the PSMSL, there is the opportunity to compute the absolute sea-level rates of rise in more locations.
Analyses of the MSL data are o ered by di erent providers such as the above-mentioned PSMSL, SONEL, sealevel.info [10], and National Oceanic and Atmospheric Administration (NOAA, [11]). While the analysis of sea level data is straightforward, the analysis of GNSS data is more troublesome, hence there is a need to use multiple providers. While SONEL is directly linked to PSMSL, it only considers a small number of GNSS antennas, 496 as per June 2018 to cover the world, with data not always up-to-date. JPL has a more extensive database of GNSS antennas, 2,822, albeit not comparable to the NGL database [5], that is by far the most extensive, with 15,277 antennas.
It is widespread to correct the relative sea-level rise trend by a GIA computation such as [1,12,13]. However, global GIA models only account in everything but a perfect way for only one of the many components of land motion, the deformation of the Earth's crust in response to changes in the polar ice caps, completely neglecting regional subsidence and crustal movement. Land subsidence is a global problem, especially relevant in the United States, where more than 45,000 km of land [14], have been directly a ected by subsidence. The principal causes are aquifer-system compaction, hydro-compaction, natural compaction, underground mining, drainage of organic soils, sinkholes, and thawing permafrost [15]. Nearly the entire East Coast of the United States, from Massachusetts and parts of Maine to Florida, is known to be a ected by subsidence [16][17][18][19].
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 estimation of the absolute vertical velocity of the tide gauge instrument. However, the GNSS monitoring of the position of antennas certainly provides a much better-quality estimation of the local absolute vertical velocity than a global glacial isostatic adjustment (GIA) model computation. 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 [2]. The need for a GNSS antenna co-located with the tide gauge is stressed by [20]. The GIA correction has been seriously questioned by [21].
If the GNSS correction is more accurate than the correction by a global GIA model that does not include any regional subsidence or crustal movements, nevertheless many technicalities limit the accuracy of the GNSS vertical velocity estimation for a speci c location, for the same antenna di erent providers may propose di erent values of the absolute vertical velocity, and nearby antennas may exhibit a strongly variable pattern of subsidence not always genuine.
In the following sections, one relative MSL result, and multiple GNSS results are proposed for every selected tide gauge location, in addition to the GIA estimations. While the subsidence rate will be based on the NGL GNSS results [5], the other estimations help to understand the uncertainties in the assumption.

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. A linear regression: returns the sea level rate of rise u as the slope B. A quadratic regression returns the acceleration a taken as 2·C. 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. The absolute rates of rise of the sea levels are then computed as v = u + w [2].

Results
Presented below are the analyses of the relative rates of rise and accelerations of the sea level in the 33 Long Term Trend (LTT) tide stations of the East Coast of North America. Figure 1 presents a map with the relative sea-level rise trends in the locations with more than 80 years of data in the PSMSL database, with the East Coast of North America in evidence. The PSMSL map only shows Halifax and Trois-Rivieres in Canada, then the US stations between Portland and Key West. Galveston, that in the Gulf of Mexico, and not strictly speaking East Coast of North America is also shown. Galveston is located in a well-known area of extreme subsidence for oil and groundwater extraction. Table 1 summarizes the relative rates of rise and accelerations for the 33 stations that are considered in the survey. u is the relative sea-level rise, w is the absolute vertical velocity at the GNSS antenna nearby the tide gauge, and v=u+w is the absolute sea-level rise. Table 1 summarizes the tide gauge results as well as the results at the nearby GNSS antennas, and the results of two GIA models. The absolute rates of rise of the sea levels are computed based Figure 1: Locations of the tide gauges with more than 80 years of data in the PSMSL database. Image reproduced modi ed after [7]. Northern-most station is Trois Riviere, Southern-most station is Key West.
on these data. The table proposes as w* the Glacial Isostatic Adjustment (GIA) vertical velocities VM2 and VM4 from [12,13]. The VM2 and VM4 data are presently available at [22]. This page has two links for Peltier's les, [23] (VM2) and [24] (VM4). The second link is now broken also in the archived versions. The data contains a known error, which was discovered in March 2012, that is however irrelevant for the paper.
Figures 2 to 16 present the details of the monthly average mean sea levels (MSL) as well as the GPS position for the more representative 15 stations, having more than 100 years of data. MSL images are from sealevel.info. GPS images are reproduced modi ed after [5].

. Saint John, N.B., Canada
The MSL trend at Saint John, N.B., Canada, Figure 2, is +2.14 mm/year with a 95% con dence interval of ±0.20 mm/year, based on MSL data from 1896/6 to 2016/12. The acceleration is -0.00631 ±0.01237 mm/yr . Saint John, N.B., Canada has no nearby GNSS stations from SONEL. JPL also has no station nearby. NGL has STJH, of absolute vertical velocity -0.177±12.814 mm/yr., SJNB, of absolute vertical velocity -0.664±1.052 mm/yr. and SJPA, of absolute vertical velocity -0.697±0.717 mm/yr. A likely estimation of the absolute vertical velocity is taken as -0.697 mm/yr., the subsidence of the NGL station of SJPA, the one with more data and still operational.
Halifax, Canada has the nearby GNSS Station from SONEL of HLFX, having absolute vertical velocity -1.11±0.18 mm/yr. The distance to the tide gauge is 3,100 m. From JPL, HLFX has absolute vertical velocity -1.07±0.213 mm/yr. From NGL, HLFX has absolute vertical velocity -0.895±0.552 mm/yr.
A likely estimation of the absolute vertical velocity is taken as -0.895 mm/yr., the NGL result for HLFX.
Charlottetown, Canada has no nearby GNSS station from SONEL. JPL also has no stations nearby. NGL has PETI, of absolute vertical velocity -1.751±1.520 mm/yr. A likely estimation of the absolute vertical velocity is taken as -1.751 mm/yr., the NGL result for PETI. Table 1: Summary of sea-level rise and subsidence results. u is the relative sea-level rise, w is the absolute vertical velocity at the GNSS antenna nearby the tide gauge, or computed with a GIA model, and v = u + w is the absolute sea-level rise. GNSS Peltier GIA
Trois-Rivieres, Canada is further south on the St. Lawrence River. SONEL and JPL have no station nearby, NGL has those stations listed before. A likely estimation of the absolute vertical velocity is taken as 0.77425 mm/yr., same as above.  [5]. c) GNSS time series for BECA. Image reproduced modi ed after [5].

. Batiscan, Canada
The MSL trend at Batiscan, Figure 8, is -1.49 mm/year with a 95% con dence interval of ±1.81 mm/year, based on MSL data from 1901/5 to 2016/12. The acceleration is -0.0376 ±0.1248 mm/yr . Batiscan, Canada is also on the St. Lawrence River, south of Quebec, north of Trois-Rivieres. In absence of any data from SONEL and JPL, a likely estimation of the absolute vertical velocity is taken as 0.77425 mm/yr., the same as above.

. Neuville, Canada
The MSL trend at Neuville, Canada, Figure 9, is +0.05 mm/year with a 95% con dence interval of ±0.70 mm/year, based on MSL data from 1914/6 to 2016/12. The acceleration is -0.0266 ±0.0520 mm/yr . Neuville, Canada is also on the St. Lawrence River, south of Quebec, north of Trois-Rivieres.
In absence of any data from SONEL and JPL, a likely estimation of the absolute vertical velocity is taken as 0.77425 mm/yr., the same as above.
Worth to mention, are GNSS antennas inland from the St. Lawrence River between Trois-Rivieres and Quebec. From NGL, QCSM has an absolute vertical velocity -0.878±3.110 mm/yr. and LAUR has absolute vertical velocity 2.367±1.021 mm/yr. Hence, more GNSS data are certainly needed to resolve this region.

. Harrington Hbr and St John's Nfld., Canada
For the other stations of Canada, of length less than 100 years, the subsidence rates in Table 1  .

Eastport and Portland, United States
Then, moving south, within the US, the rst station with more than 100 years of data is Portland. Before Portland, there is Eastport. Eastport, ME, United States has the nearby GNSS Station from SONEL EPRT, of absolute vertical velocity 0.28±0.25 mm/yr. The distance to the tide gauge is 853 m. NGL has EPRT, of absolute vertical velocity 0.103±0.863 mm/yr. A likely estimation of the absolute vertical velocity is taken as 0.103 mm/yr., the NGL result for EPRT. This is the value used in Table 1.
The MSL trend at Portland, ME, United States, Figure  10, is +1.87 mm/year with a 95% con dence interval of ±0. 15   .

Boston, Woods Hole, Newport, and Kings Pt/Willets Pt, United States
In between Portland and NY, there are several tide gauges of length less than 100 years. The subsidence rates in Table  1

. The Battery, United States
The MSL trend at The Battery, NY, United States, Figure 11, is +2.85 mm/year with a 95% con dence interval of ±0.09 mm/year, based on MSL data from 1856/1 to 2018/3. The acceleration is 0.00849 ±0.00388 mm/yr . The Battery, NY, United States has the nearby GNSS Stations from SONEL of NYBR, with no data, and NYBP, of absolute vertical velocity -2.12±0.62 mm/yr. NYBP is practically co-located at a distance to the tide gauge of only 49 m. JPL has no data. NGL has NYBP, of absolute vertical velocity -2.152±0.969 mm/yr.; and NYBR, of absolute vertical velocity -1.085±1.021 mm/yr. As NYBP is co-located with the tide gauge, a likely estimation of the absolute vertical velocity is taken as -2.152 mm/yr., the NGL result for NYBP. .

Sandy Hook, United States
The subsidence rate in Table 1  .

Atlantic City, United States
The MSL trend at Atlantic City, NJ, United States, Figure 12, is +4.08 mm/year with a 95% con dence interval of ±0 .

Philadelphia, United States
The MSL trend at Philadelphia, PA, United States, Figure  13, is +2.94 mm/year with a 95% con dence interval of ±0.19 mm/year, based on MSL data from 1900/7 to 2017/12. The acceleration is 0.01607 ±0.01221 mm/yr . Philadelphia, PA, United States has the nearby GNSS Station from SONEL of PAPH, of absolute vertical velocity -0.53±0.38 mm/yr. The distance to the tide gauge is 9,390 m. NGL has PAPH, of absolute vertical velocity -0.533±1.081 mm/yr. A likely estimation of the absolute vertical velocity is taken as -0.533 mm/yr., the NGL result for PAPH.

Baltimore, United States
The MSL trend in Baltimore, MD, United States, Figure 14, is +3.15 mm/year with a 95% con dence interval of ±0.13 mm/year, based on MSL data from 1902/6 to 2017/12. The acceleration is 0.00382 ±0.00852 mm/yr . .

Annapolis, Solomon's Island, Washington, DC, Sewells Point, Wilmington, Charleston, and Fort Pulaski, United States
South of Baltimore and north of Fernandina Beach there are other tide gauges of length less than 100 years. In these tide gauges, the subsidence rates shown in Table 1 are obtained as follows.
Annapolis, MD, United States has the nearby GNSS Stations from SONEL of ANP6, of no data, USNA, of absolute vertical velocity -0.63±0.53 mm/yr. ; LOYF, of no data, ANP1, of absolute vertical velocity -1.03±0.46 mm/yr.; and ANP5, of absolute vertical velocity -2.46±0.37 mm/yr. USNA is co-located with the tide gauge, with a distance to the tide gauge of 37 m. NGL has, in addition to the others, USNA, of absolute vertical velocity -1.551±1.231 mm/yr. A likely estimation of the absolute vertical velocity is taken as -1.551 mm/yr., the NGL result for USNA.  .

Mayport, United States
South of Fernandina Beach, the tide gauge record in Mayport is less than 100 years long. The subsidence in Table 1 is obtained as follows. Mayport, FL, United States has no nearby stations from SONEL. No nearby stations are provided by JPL. Relatively far, NGL has the above mentioned JXVL, of absolute vertical velocity -0.164±1.398 mm/yr. A likely estimation of the absolute vertical velocity is taken as -0.164 mm/yr.

. Key West, United States
Finally, we reach Key West, the southernmost point of the continental United States. The MSL trend at Key West, FL, United States, Figure 16, is +2. 42

Discussion
In the 33 LTT stations along the East Coast of North America, the average relative rate of rise is 2.22 mm/yr. subjected to a small, positive acceleration of +0.0027 mm/yr . The average relative rate of rise of the 11 stations in Canada is 0.61 mm/yr. subjected to a negative acceleration of -0.0133 mm/yr while theaverage relative rate of rise of the 22 stations of the United States is 3.02 mm/yr. subjected to a positive acceleration of +0.0108 mm/yr . Excessive groundwater withdrawal-induced subsidence is much stronger for the East coast of the United States than Canada, The acceleration result is consistent with other global and regional estimations from LTT stations such as [25] to [28] 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 . The Holgate's nine excellent tide gauge records of sea-level measurements have average acceleration +0.0029±0.0118 mm/yr . The NOAA's 42 U.S. long term trend tide stations of 2011 have average acceleration +0.0025±0.0308 mm/yr . The California-8 long term trend tide stations have average acceleration +0.0014±0.0266 mm/yr . The LTT stations of the East 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.
In addition to Table 1, also Figure 17 presents a summary of the sea level and GNSS results for the LTT stations of the East coast of North America. u is the relative sealevel rise, w is the absolute vertical velocity at the GNSS antenna nearby the tide gauge, and v=u+w is the absolute sea-level rise.
In the 11 stations of Canada, of average relative sealevel rise 0.61 mm/yr., the average absolute velocity of the tide gauge instrument as guessed from the GNSS data is 0.43 mm/yr. translating in an absolute sea-level rise of +1.04 mm/yr. The average absolute velocity of the tide gauge instrument from GIA VM2 and VM4 is 0.02 mm/yr. and -0.16 mm/yr. respectively. The absolute rate of rise of the sea level from GIA VM2 is 0.63 mm/yr. In the 22 stations of the US, of average relative sealevel rise 3.02 mm/yr., the average absolute velocity of the tide gauge instrument as guessed from the GNSS data is -0.96 mm/yr. translating in an absolute sea-level rise of 2.06 mm/yr. The average absolute velocity of the tide gauge instrument from GIA VM2 and VM4 is -1.15 mm/yr. and -0.42 Figure 18: Selected, known areas of land subsidence, owing primarily or secondarily to groundwater or oil and gas abstractions, in the 48 conterminous United States, and associated aquifer systems. Image reproduced modi ed from [29]. mm/yr. respectively. The absolute rate of rise of the sea level from GIA VM2 is 1.88 mm/yr.
In all the 33 stations, of average relative sea-level rise 2.22 mm/yr., the average absolute velocity of the tide gauge instrument as guessed from the GNSS data is -0.50 mm/yr. translating in an absolute sea-level rise of 1.72 mm/yr. The average absolute velocity of the tide gauge instrument from GIA VM2 and VM4 is -0.76 mm/yr. and -0.33 mm/yr. respectively. The absolute rate of rise of the sea level from GIA VM2 is 1.46 mm/yr.
The subsidence along the East Coast of the United States is very well-known [14,29,30]. Hence, it should not be a surprise if along the East Coast of the United States the relative rates of rise of the sea level are generally higher than along the West Coast of the United States, where apart from southern California, subsidence is less relevant.

Conclusions
The GNSS monitoring of the position of antennas, is superior to GIA model computations, to assess vertical land velocities in general, and vertical movements of tide gauge instruments in particular. However, the technique still suffers from major uncertainties, as demonstrated by the differences between the estimates from di erent providers for the same antennas that are often larger than the trend. 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 33 LTT stations along the East Coast of North America, the average relative rate of rise is 2.22 mm/yr. subjected to a small, positive acceleration of +0.0028 mm/yr . The average relative rate of rise of the 11 stations in Canada is 0.61 mm/yr. subjected to a negative acceleration of -0.0133 mm/yr . The average relative rate of rise of the 22 stations of the United States is 3.02 mm/yr. subjected to a positive acceleration of +0.0108 mm/yr . The average absolute velocity of the 33 tide gauge instruments from the GNSS data is -0.52 mm/yr. The average absolute sea-level rise is 1.70 mm/yr. The average absolute velocity of the 33 tide gauge instruments from GIA VM2 and VM4 is -0.76 mm/yr. and -0.33 mm/yr. respectively translating in averages absolute sea-level rise of 1.46 mm/yr. and 1.89 mm/yr. from GIA VM2 and VM4, respectively.
The result here obtained is compatible with a global temperature recovery from the low temperatures of the Little Ice Age (LIA) plus the e ect of subsidence. This paper adds to the many contributions, such as [25] to [28], [31] to [79] that demonstrate the world sea levels are rising slowly and with only a small acceleration component.