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Appendix A
Vertical Land Motion and Sea-Level Data
Along the West Coast of the United States
A
s summarized in Chapter 4 ("Analysis of West signal. The committee used CGPS data to estimate and
Coast Tide Gage Records"), the committee remove the vertical land motion component from the
determined rates of historical sea-level change sea levels recorded by west coast tide gages. The loca-
along the California, Oregon, and Washington coasts tions of the tide gages and CGPS stations analyzed in
using 12 tide gages. The rates were then corrected for this report are shown in Figure A.1.
vertical land motion and atmospheric pressure effects Several published GPS solutions are available for
to compare sea-level rise along the west coast of the estimating west coast vertical land motions, each of
United States with the global mean sea-level rise. which uses a different number of stations, timespan,
Details of these analyses are given below.
and/or processing software (e.g., Donnellan et al.,
1993a,b; Dong et al., 1998; Argus et al., 1999, 2005;
VERTICAL LAND MOTION FROM Bennett et al., 1999; Argus and Gordon, 2001; Spinler
CONTINUOUS GPS et al., 2010). The committee used the Scripps Orbit and
Permanent Array Center (SOPAC) velocity model,2
Tide gages are generally corrected for vertical which is a routinely updated, publicly available solu-
land motions using glacial isostatic adjustment (GIA) tion with the longest time span for each station as well
models. However, GIA models capture only a small
as the greatest GPS station density for the U.S. west
component of the total vertical land motion in coastal coast. The SOPAC processing details are described
areas that are tectonically active or undergoing subsid- in Nikolaidis (2002). Briefly, GAMIT and GLOBK
ence or uplift associated with sediment compaction software (Dong et al., 1998) are used to calculate daily
and/or fluid withdrawal or recharge. The Global site positions, which are input to the velocity estimation
Positioning System (GPS) began to be used to adjust model. Using the entire time series for a specific site,
tide gage data for vertical land motion around 1997 the model accounts for offsets, linear velocity, annual
(Ashkenazi et al., 1993; Nerem et al., 1997; Zerbini, and semi-annual fluctuations (for stations with at least
1997). Continuous GPS (CGPS) records along the 2 years and 1 year of data, respectively), and post-
U.S. west coast have been available since the early seismic relaxation. Noise analysis (Williams, 2003)
1990s, although significantly more stations have been using white noise plus flicker noise covariances provides
installed since 2003 as part of the National Science realistic uncertainty estimates.
Foundation's Plate Boundary Observatory.1 Continu- A vertical land motion value was assigned to each
ous GPS solutions allow vertical land motions to be tide gage site by taking the closest CGPS station with a
estimated from shorter temporal records and with more velocity estimate within 15 km from the tide gage. The
confidence than episodic sampling of the land motion
2See
154 APPENDIX A
-128° -124° -120° -116°
Friday Harbor
Neah Bay SC02
NEAH
48°
Seattle
SEAT
Astoria
TPW2
44°
Crescent City PTSG
40°
San Francisco TIBB
P224
Alameda
36°
Port San Luis
P524
Santa Monica WRHS
Los Angeles VTIS
La Jolla SIO3
San Diego
P475
32°
FIGURE A.1 Map showing names and locations of the 12 tide gages and CGPS stations analyzed in this report.
APPENDIX A 155
15-km value is somewhat arbitrary, but it is similar to ability in vertical land motion between a CGPS station
the distance threshold typically used in previous joint and the tide gage, and eref is the error associated with the
GPS and tide gage analyses (e.g., Mazzotti et al., 2007; definition of the GPS reference frame. Of these error
Wöppelmann et al., 2007). The committee's estimated sources, ev is well defined (Nikolaidis, 2002; Williams
rates of vertical land motions are given in Table A.1. et al., 2004) and is taken to be the vertical land motion
Positive values of vertical land motion mean that the error associated with the nearest CGPS station to the
land is rising. Table A.1 also reports the standard de- tide gage (and fifth and sixth columns of Table A.1).
viation of vertical land motion within a 15-km radius Error in the spatial variability of vertical land
of each tide gage station. This value was used in the motion between a CGPS station and a tide gage, esv,
uncertainty estimate described below. is locally variable and cannot be determined without
To test the importance of the CGPS solution on more detailed studies. For example, Brooks et al. (2007)
the calculated rate of vertical land motion, the com- reported a variation of ~ ± 3 mm yr-1 for esv in the Los
mittee compared the SOPAC-derived rates with rates Angeles Basin. To estimate esv, the committee used
published in the literature. Although most published the standard deviation of vertical land motion within
reports for coastal areas south of the Mendocino Triple a 15-km radius of each tide gage (right two columns
Junction present only horizontal rates (e.g., Spinler et in Table A.1). Using this value, rather than the formal
al., 2010), the SOPAC rates for Cascadia are similar to error estimate associated with any given GPS station's
vertical rates for the same sites published in Mazzotti velocity, seemed justified, given the potential for signifi-
et al. (2008). Mazzotti et al. (2008) estimated rates cant variability in vertical land motion at the km scale.
using different GPS processing software (BERNESE) For instance, the Santa Monica tide gage has seven
than that used by SOPAC. The correspondence in rates CGPS stations within a 15-km radius, with vertical
produced by two independent approaches suggests that land motion estimates ranging from -1.5 mm yr-1 to
the committee's results are reasonable. 1.8 mm yr-1. The nearest CGPS station, WRHS, has the
minimum vertical land motion estimate (-1.5 mm yr-1)
Uncertainty in Vertical Land Motion and may be the most appropriate value for the correc-
tion. However, confidence in this value is diminished
The estimated error in the vertical land motion by the local spatial variability, which is reflected in the
adjustment to the tide gage records can be expressed as 15-km standard deviation value (Table A.1).
eTG = ev + esv + eref, where ev is the error in the velocity The reference frame error (eref ) is a classical geo-
estimation, esv is the error associated with spatial vari- detic problem (Strang and Borre, 1997). The SOPAC
TABLE A.1 Parameters for Vertical Land Motion Correction
Vertical Land Motion, Vertical Land Motion,
Nearest Station 15-km Radiusa
Distance Rate Rate
Tide Gage Nearest CGPS Station Start Date (km) (mm yr-1) Error (1) (mm yr-1) Error (1)
Friday Harbor SC02 2001.860 0.70 0.90 0.70
Neah Bay NEAH 1996.000 7.71 3.00 0.40
Seattle SEAT 1996.000 6.25 0.20 0.50 -1.10 0.94
Astoria TPW2 2000.247 1.08 0.60 0.00 1.20 0.40
Crescent City PTSG 1999.820 5.85 2.60 0.40
San Francisco TIBB 1994.460 10.23 -1.40 0.50 -1.44 1.97
Alameda P224 2005.174 12.90 -0.20 0.60 -1.58 1.20
Port San Luis P524 2007.048 14.50 1.70 0.30
Santa Monica WRHS 1999.770 9.36 -1.50 0.60 -0.01 1.34
Los Angeles VTIS 1998.938 2.54 -0.50 0.50 -0.27 2.34
La Jolla SIO3 1993.522 0.26 2.10 0.50 0.73 1.11
San Diego P475 2007.601 9.17 -3.00 0.20 -4.50 0.81
a Rates and errors are not reported for 15-km areas with only 1 CGPS station.
156 APPENDIX A
TABLE A.2 Tide Gage Records from PSMSL, Corrected for Atmospheric Pressure and Vertical Land Motion
Trend (mm yr-1) Confidence Limit Trend (mm yr-1)
With IB With IB
Original With IB and GPS Upper Lower and GIA
Tide Gage Latitude Longitude Period Record Adjustment Adjustment 95% 95% Adjustment
Friday Harbor 48.550 -123.000 19342008 +1.04 +1.14 +2.04 +3.44 +0.64 +1.31
Neah Bay 48.367 -124.617 19342008 -1.77 -1.65 +1.35 +2.22 +0.48 -1.33
Seattle 47.760 -122.333 19002008 +2.01 +2.10 +2.30 +3.29 +1.31 +1.67
Astoria 46.217 -123.767 19252008 -0.38 -0.30 +0.30 +0.61 -0.01 -1.37
Crescent City 41.750 -124.200 19332008 -0.73 -0.65 +1.95 +2.78 +1.12 -0.87
San Franciscoa 37.800 -122.467 19002008 +1.92 +1.98 +0.58 +0.69 +0.47 +1.99
Alameda 37.767 -122.300 19392008 +0.70 +0.82 +0.62 +1.82 -0.58 +0.85
Port San Luis 35.167 -120.750 19452008 +0.68 +0.76 +2.46 +3.10 +1.82 +0.68
Santa Monica 34.017 -118.500 19332008 +1.41 +1.44 -0.09 +0.72 -0.90 +1.43
Los Angeles 33.717 -118.267 19232008 +0.84 +0.83 +0.33 +1.32 -0.66 +0.80
La Jolla 32.867 -117.250 19242008 +2.08 +2.07 +4.17 +5.17 +3.17 +2.02
San Diego 32.717 -117.167 19062008 +2.04 +2.07 -0.96 +0.03 -1.95 +2.01
NOTE: IB = inverse barometer.
a Although the San Francisco record starts in 1854, the IB correction starts at 1900.
velocity solution uses the International Terrestrial the difficulty of estimating eref in an absolute sense, the
Reference Frame's ITRF2005 realization,3 one of the committee adopted a conservative value of ± 1 mm yr-1
most accurate and rigorously constrained versions of for eref . The committee's error estimates for the vertical
a global reference frame. ITRF2005 was developed land motion correction to the tide gages are given in
from a combination of space geodetic observations Table A.2.
from four independent platforms: GPS, Very Long
Baseline Interferometry, Satellite Laser Ranging, and ANALYSIS OF SEA-LEVEL TREND FROM
Doppler Orbit Determination and Radio-Positioning TIDE GAGES
Integrated on Satellite. ITRF2005 is an improvement
over previous frame realizations because it uses better Sea-level records are archived at the Permanent
antenna phase center models and its use of time series Service for Mean Sea Level (PSMSL).4 PSMSL re-
enables treatment of nonlinear and discontinuous be- duces the data reported from each tide gage to monthly
havior. A detailed description of the ITRF2005 realiza- mean values and adjusts them to a common datum to
tion appears in Altamimi et al. (2007). produce a revised local reference dataset (Woodworth
Altamimi et al. (2007) suggested that the dis- and Player, 2003). The sea-level data used in this report
agreement in origin definition between ITRF2005 are all revised local reference datasets. PSMSL gives sea
and ITRF2000 (the previous frame realization) could level in mm and, to avoid negative numbers, adds 7,000
be used as a metric of the origin accuracy. They re- mm to each monthly mean.
ported a translation misfit of 0.1 mm, 0.8 mm, and The committee examined the sea-level records from
5.8 mm along the x-, y-, and z-axes, respectively, with the 28 PSMSL tide gages along the west coast of the
a formal error of 0.3 mm for each component. The United States (Table A.2). For its analysis, the commit-
misfit of translation rate was reported as 0.2 mm yr-1, tee chose 12 tide gages that are currently operating and
0.1 mm yr-1, and 1.8 mm yr-1, with a formal error of have records at least 60 years long. Shorter records are
0.3 mm yr-1 (Altamimi et al., 2007). Mazzotti et al. subject to decadal bias (Box A.1; see also "Tide Gage
(2007), using ITRF2000, estimated that their vertical Measurements" in Chapter 2) and were not analyzed.
land motion values could contain a reference frame Records with gaps in the data such as Santa Monica
bias of -0.50.8 mm yr-1. Given these two results and were simply treated as nonuniformly spaced in time.
3 See . 4 See .
APPENDIX A 157
BOX A.1
Effect of Record Length on Sea-Level Trend
As noted in Chapter 2, different lengths of record yield different sea-level trends because of decadal variability. The committee investigated the
effect of record length on the rate of sea-level rise by fitting a linear trend over varying lengths of record from the San Francisco tide gage. The top
figure shows the trends based on a starting year of 1900 and using longer and longer record lengths. The resulting rates of sea-level rise range from
1.12 mm yr-1 to 2.10 mm yr-1. The bottom figure shows the reverse analysis, fixing the fit line at 2009 and adding progressively older data. For this case,
rates of sea-level rise range from -0.05 mm yr-1 to 2.66 mm yr-1, and the recent (19801990) downtrend in sea level noted in Bromirski et al. (2011) can
be seen. The wide range in sea-level trends depending on the starting time and the length of the record also shows the difficulty of describing a complex
time-varying signal with a simple linear relationship.
FIGURE Sea-level trends for the San Francisco tide gage data as a function of data length. (Top) Straight line fits from 1900.
(Bottom) Straight line fits from 2009.
158 APPENDIX A
Although this approach has little or no effect on the Sea-level pressure data were obtained from the
estimated rate of sea-level rise, it slightly reduces the un- National Oceanic and Atmospheric Administration
certainty (variability) when the gaps are relatively long. Earth System Research Laboratory (NOAA ESRL)
The committee began the analysis of the PSMSL 20th Century Reanalysis V2 database.5 These data are
data by removing the 7,000 mm offset, thus reintroduc- available on a 2° × 2° global grid from 1871 through
ing negative values. Long-term trends for the original 2008. The sea-level pressure for each tide gage (P(t))
tide gage data were obtained by fitting a straight line was extracted from the nearest 2° × 2° box. The mean
to the monthly average values for each record using a ocean surface pressure for each month (P(t)ref ) was ob-
least squares method expressed as yt = mxt + b, where tained by averaging over all 2° × 2° boxes. The results of
yt (in mm) is the monthly mean tide level at time xt, the IB adjustment are shown in Table A.2. The effect
where t = 1, 2,..., N, and N equals the total number of of the adjustment is relatively small, changing the slope
observations in the record. The slope m, which repre- by less than 10 percent in the majority of cases. The
sents the relative sea-level rise, is given in mm yr-1, and changes appear to be smallest in southern California.
b is the y-intercept in mm. The linear sea-level trends
(m) were determined using the entire record length of Vertical Land Motion
each tide gage.
The tide gage records were corrected for local site
Adjustments to the Original Tide Gage Trends motion using GPS data (see "Vertical Land Motion
from Continuous GPS" above). The CGPS rates of
The committee adjusted the original tide gage vertical land motion were simply added to the tide gage
trends to remove the effects of atmospheric pressure rate of relative sea-level rise (the slope m) to obtain a
and vertical land motions, as described below. new local motion-adjusted record. The GPS data ex-
tend back in time only one to two decades, while some
Atmospheric Pressure tide gage records are more than 100 years long. For
correction purposes, the committee made the crucial
In many studies of sea-level rise, tide gage records assumption that the GPS adjustment remains constant
are not adjusted for the barotropic response of the ocean over the entire length of the sea-level record. This as-
due to variations in atmospheric pressure. However, sumption is more likely valid where the vertical land
observations suggest that an ocean response to atmo- motion is dominated by GIA, but it is open to question
spheric pressure loads is expected everywhere except the where subsidence or uplift are the primary geophysical
tropics and western boundary current extension regions causes. In most cases, the GPS adjustments are sig-
(Wunsch and Stammer, 1997). The so-called inverse nificantly larger than the GIA adjustments, confirming
barometer (IB) adjustment is a good approximation of the importance of tectonics and subsidence to relative
the barotropic response of sea level. sea-level rise in the area (Table A.2).
The committee adjusted the 12 tide gage records The GPS adjustment is generally larger than the IB
for the inverse barometer effect following the proce- adjustment. The magnitude of the two adjustments var-
dures of Ponte et al. (1991). The adjustment can be ies among tide gages, with relatively large changes to the
expressed as (P(t) - P(t)ref )/g, where is the density original trend in a few places due primarily to the GPS
of sea water and g is the acceleration of gravity. The adjustment. For example, these corrections changed the
term -1/g is a scale factor that converts local air pres- slopes from -1.77 mm yr-1 to +1.35 mm yr-1 at Neah
sure anomalies to water level equivalent with a value of Bay, from -0.73 mm yr-1 to +1.95 mm yr-1 in Crescent
-9.948 mm mb-1. P(t) represents the monthly averaged City, and from +2.04 mm yr-1 to -0.96 mm yr-1 at San
sea-level pressure for a specific tide gage and month at Diego (Table A.2). The large change at San Diego
time t, and P(t)ref is the mean surface pressure taken underscores the importance of small-scale spatial vari-
over the global ocean for the same month. This adjust- ability in vertical motion in the region.
ment is then subtracted from the monthly averaged sea
5 See
APPENDIX A 159
Results a given value and the line of regression. For a more
detailed explanation, see Huber (1981). The process for
Figure A.2 shows the distributions of slopes for the deciding what weighting to use is somewhat subjective,
original tide gage data, the data with IB adjustments, but tests using several different weight functions did
and the data with IB and GPS adjustments for the 12 not significantly affect the results. The calculated slopes
tide gages. The mean value of all slopes is 0.82 mm yr-1 seldom varied by more than ± 10 percent.
for the original sea-level records, 0.88 mm yr-1 with To examine the variation in sea-level trends along
the IB adjustment, and 1.25 mm yr-1 with the IB and the coastline, a weighted regression was applied to all
GPS adjustments. The standard deviation increases 12 gages. The linear trend in the weighted regression
from about 1.23 mm yr-1 to 1.39 mm yr-1 when the for the entire coast shows a significant increase in
GPS adjustment is included. This slight increase sug- slope from south to north (Figure A.3, solid red line),
gests that the local variability in vertical motion is often whereas the original data (dashed red line) show a de-
significant and uncorrelated with the vertical motion crease from south to north. The change in trend reflects
experienced by the tide gage. It also may indicate that vertical land motion along the coast.
vertical land motions over the past one or two decades
are not representative of the ground motion over the
Uncertainty in Adjusted Sea-Level Trends
lifetime of the tide gage.
To suppress the influence of possible outliers, the The 95 percent confidence limits in the sea-level
slopes were plotted as a function of latitude then fit with trends were calculated using measures of uncertainty
robust or weighted lines of regression (solid red line in for the original records and the GPS adjustments. For
Figure A.3). In weighted regression, values that lie far data where the individual observations are statistically
from the line of regression are given less weight than independent, confidence limits are calculated using
values that lie closer to the line. The weighted regres- the student's t distribution. However, with time series,
sion employs iteratively reweighted least squares with the observations are generally not independent, so the
a selection of weighting functions depending on what equivalent number of independent observations or
form and how much weighting is desired (Holland and degrees-of-freedom (DOFs) is often far less than the
Welsch, 1977). The form of weighting chosen was the total number of observations in the original time series.
Welsch weight function. The weights, w(r), are given For DOFs greater than 120, the t values do not change
by exp(-r 2), where r is related to the distance between and no adjustment for serial correlation is required.
FIGURE A.2 Distributions of slopes for the original relative sea-level data (left), with inverse barometer (IB) corrections (center), and
with IB plus GPS corrections (right).
160 APPENDIX A
FIGURE A.3 Slopes of linear trends for tide gage records and weighted lines of regression. Slopes for original tide gage records
(stars), for IB adjusted records (+), and for IB and GPS adjusted records (o) are shown as a function of latitude from north to south,
together with weighted linear trends for the entire coast for the original data (dotted red line) and for the GPS-adjusted data (solid
red line).
Several approaches are used to estimate the DOFs uncertainty is equal to the square root of the sum of
when the data are serially correlated, as they are in this the variances associated with each. Multiplying this
case. The committee used the Mitchell et al. (1966) value by 1.96 yields the 95 percent confidence limits.
approach to calculate the DOFs for each record, and The upper and lower confidence limits are included in
found that the DOFs were greater than 120 in all cases. Table A.2. In some cases, the uncertainties associated
Calculating the 95 percent confidence limits reduces to with the GPS data far exceed those associated with the
the slope ± 1.96 (from the t table) times the square root tide gage data.
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