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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, B02404, doi:10.1029/2008JB006155, 2010

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Tide gauge and GPS measurements of crustal motion and sea level rise along the eastern margin of Adria G. Buble,1 R. A. Bennett,1 and S. Hreinsdo´ttir1 Received 14 October 2008; revised 30 August 2009; accepted 16 September 2009; published 6 February 2010.

[1] We use observations from tide gauges and colocated continuous GPS (CGPS) stations

to investigate crustal deformation and sea level changes along the eastern margin of the Adriatic Sea. We develop a new method to separate common-mode relative sea level from spatially variable signals. Precise vertical crustal motions determined by CGPS allow us to further separate relative sea level into absolute sea level changes and crustal motions with respect to a local Central Mediterranean–fixed GPS-defined reference frame. From the tide gauge data, we find fairly uniform relative sea level rise along the coast, with mean rate of 0.84 ± 0.04 mm/yr and weighted RMS variation about this mean of 0.2 mm/yr. This rate is a factor of 2–4 lower than estimates for global average sea level rise. In contrast, vertical motion of coastal rocks determined by CGPS vary appreciably from an average of 1.7 ± 0.4 mm/yr in southern Adria to 0.0 ± 0.4 mm/yr in northern Adria. This difference in crustal motion between the northern and southern regions is independent of our ability to separate sea level from crustal motion, and may be explained by crustal strain associated with an active thrust fault accommodating southern Adria microplate convergence with Eurasia. Enigmatically, the combination of tide gauge and CGPS measurements shows that absolute sea level relative to the GPS-determined reference frame varies by as much as 1.8 mm/yr along the Croatian coast in such a way that the relative sea level remains roughly constant. There are several potential explanations for this result deriving from ocean, atmosphere, and solid Earth dynamics. Citation: Buble, G., R. A. Bennett, and S. Hreinsdo´ttir (2010), Tide gauge and GPS measurements of crustal motion and sea level rise along the eastern margin of Adria, J. Geophys. Res., 115, B02404, doi:10.1029/2008JB006155.

1. Introduction [2] Knowledge of relative sea level trends along the eastern coast of the Adriatic Sea (Figure 1) is critical for studies of coastal hazards, sea level history, and vertical crustal motion associated with local tectonics and other geodynamic processes. The absence of paleobeaches along this coast clearly indicates that the sea surface has risen relative to the land surface since the last glacial maximum. However, many fundamental issues remain unresolved. For example, past estimates for absolute sea level rise in the Adriatic (0.4 – 1.4 mm/yr) [Woodworth, 2003] are lower than the global average recorded by tide gauges (1.8 ± 0.1 mm/yr) [Douglas, 1997] and satellite altimetry (2.8 ± 0.4 mm/yr) [Cazenave and Nerem, 2004]. Moreover, most of the past estimates for Adria tide gauge trends [e.g., Orlic´ and Pasˇaric´, 2000; Woodworth, 2003; Marcos and Tsimplis, 2008] differ from one station to another with an RMS variation of 0.8 mm/yr about the mean. Differences among reported trend estimates and estimates for sea level rise may signify a number of causes, including a natural response to redistribution of water and ice mass over the Earth’s surface; 1 Department of Geosciences, University of Arizona, Tucson, Arizona, USA.

Copyright 2010 by the American Geophysical Union. 0148-0227/10/2008JB006155$09.00

sea level accelerations through time; motions of tide gauge benchmarks associated with local tectonics and other regionalscale geodynamic processes; or rate biases associated with measurement technique (e.g., calibration errors, imperfect spatial, and temporal sampling), or interannual and decadal ocean variability. [3] These issues have been difficult to resolve for two main reasons. First, tide gauges record sea level relative to benchmarks attached to crustal rocks. Crustal motions along the eastern Adria coast and other coastlines around the world are uncertain. For example, existing models for the isostatic response of coastal regions to the loss of the Pleistocene ice sheets predict a wide range of crustal motions. For the Adriatic coastal region, some isostatic response models predict subsidence of 0.3 – 0.5 mm/yr [Lambeck and Johnston, 1995; DiDonato et al., 1999; Stocchi and Spada, 2009], whereas others predict uplift of 0.2– 0.3 mm/yr [Tushingham and Peltier, 1989; Douglas, 1991]. Furthermore, vertical crustal motion associated with interseismic crustal deformation resulting from active tectonic processes is typically poorly known. Tectonic deformation along the eastern margin of Adria has been modeled only along the southern coastal region and only horizontal deformation was considered [Bennett et al., 2008]. Second, unbiased estimation of secular (long term), spatially averaged relative sea level trends is hampered by the generally unpredictable nature and large amplitude of interannual

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BUBLE ET AL.: TIDE GAUGE RECORD IN EASTERN ADRIA

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Figure 1. Tectonic settings of the Adria microplate (modified from Bennett et al. [2008]). Some of the boundaries of Adria are uncertain (dashed line and question marks). The general motion of Adria relative to Eurasia is indicated by the velocity vectors. Abbreviations: Al, Alps; Alb, Albanides; Ap, Apennines; AS, Adriatic Sea; Di, Dinarides; Du, Dubrovnik; Eu, Eurasia; Is, Istrian peninsula; N Ad, North Adria; Nu, Nubia; Ri, Rimini; S Ad, South Adria; SO, Strait of Otranto. Locations of CGPS stations that define the Central Mediterranean vertical reference frame are shown with black squares, and CGPS stations on the eastern coast of Adria are shown with circles. variations in sea level caused by changes in atmospheric pressure, water chemistry and temperature, and other variables [e.g., Tsimplis and Baker, 2000]. Biases are likely to be largest for short time series, such as provided by satellite altimetry data, when restricted to geographically limited regions such as Adria, wherein interannual and decadal variations may be spatially coherent, and thus, do not cancel by forming spatial averages. [4] While temporal variability of sea level is likely to be related to oceanic and climatic processes [e.g., Tsimplis and Baker, 2000], spatial variations may reflect differences in absolute sea level rise and vertical motion of crustal rocks associated with geodynamic processes. Sea level rise is known to vary spatially as a result of the redistribution of mass within the Earth system [e.g., Mitrovica et al., 2001], but these variations are generally assumed to be of long wavelength (>1000 km). However, shorter-wavelength spatial variations in absolute sea level rise may be caused by secular changes in sea water temperature, salinity, and other conditions between the south and north Adria subbasins, either natural or antrophogenic in origin. If absolute sea level variability has long spatial wavelength relative to the pattern of vertical crustal motion and tide gauge station spacing, one might expect to separate the spatially variable trends associated with tectonics from the broader-scale variations associated with absolute sea level using tide gauges and geodynamic models alone. As we show in

section 5, absolute sea level rise may vary appreciably across short distances. In this case, separation of sea level from crustal motion requires independent measurement of crustal motion, such as provided by continuous GPS (CGPS) [e.g., Wo¨ppelmann et al., 2007; Kierulf et al., 2008; Teferle et al., 2008]. [5] The Adriatic tide gauge record (Figure 2) is an ideal data set to investigate sea level and crustal deformation contributions to relative sea level trends. Active tectonic processes affect a broad segment of the eastern Adria margin based on seismicity, geodetic measurements of horizontal crustal motion, and other indicators [Anderson, 1987; Benac et al., 2004; Bennett et al., 2008]. The Adriatic Sea is a relatively small and isolated body of water in which spatial variation in absolute sea level rate might be expected to be negligible. Moreover, tide gauges along the eastern coast of the Adriatic Sea provide one of the best records of relative sea level change in the world, both in terms of the spatial density of tide gauges and their duration of operation. There are six tide gauges suitable for this study, with average station spacing of 100 km (Figure 2). Most of these instruments have been in operation for more than 50 years, sufficient to precisely determine long-term trends. This favorable spatial and temporal sampling should allow for the separation of common-mode relative sea level trend from spatially variable relative sea level. Moreover, there are precise CGPS stations colocated with three of the tide

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Figure 2. Locations of tide gauges on eastern margin of Adria. Tide gauges used in this study (black circles) and other tide gauges unsuitable for this study due to insufficient data, or vertical motion not related to tectonic processes (hollow circles). Notation: 1, Venezia; 2, Koper; 3, Rijeka; 4, Losinj; 5, Zadar; 6, Hvar; 7, Sucuraj; 8, Bar.

gauges. When combined with data from large-scale networks of CGPS stations throughout Europe and globally, the Adria CGPS stations can provide precise measurements of vertical crustal motion relative to an external frame of reference. Thus, the combined GPS and tide gauge data set provides a means of assessing both crustal motion and sea level relative to a GPS-defined external reference frame. [6] The primary objective of this paper is to use the Adriatic tide gauge and CGPS data sets to investigate the relative contributions of sea level and crustal motion to relative sea level trends. We review previously published material on Adria sea level, tectonics, and glacial isostatic adjustment (GIA) in section 2. In section 3, we provide a detailed description of the tide gauge and CGPS data sets. We develop methods to analyze the tide gauge record and compare our tide gauge results with CGPS measurements of vertical crustal motion in section 4. We consider possible geophysical interpretations of the results in section 5.

2. Background [7] The Adriatic Sea extends northwest from 40.0°N to 45.5°N latitude. It is relatively shallow, especially in the north between the Istria peninsula (Croatia) and Rimini (Italy), where the depth does not exceed 46 m. A wellmarked depression that exceeds 180 m in depth is located in the middle of the northern half of the sea. The deepest part of the sea is located south of Dubrovnik, where depths exceed 1300 m. The mean depth of the Adriatic Sea is estimated at 240 m. The sea reaches a maximum width of about 160 km, while in the Strait of Otranto, it narrows to 85 km width. The Adriatic is a ‘‘closed’’ sea, with inflow

coming only through the Strait of Otranto. The effect of atmospheric pressure variations on relative sea level was investigated by Orlic´ and Pasˇaric´ [2000], who found that year-to-year fluctuations of sea level could be correlated with the simultaneous variations of air pressure, a 1-mbar increase (decrease) of atmospheric pressure resulting in a 1.8– 2.0 cm lowering (rising) of sea surface. [8] Relative sea level change for the Adriatic has been investigated using a variety of techniques from which no general agreement has arisen. TOPEX/POSEIDON and Jason-1 satellite altimetry missions provide an estimate of 7 mm/yr, representing secular motion averaged over the time period of 1993 –2002 [Fenoglio-Marc et al., 2004; Garcia et al., 2007; Mangiarotti, 2007]. Although altimetry provides relatively high spatial resolution, the rates obtained for this short duration of time may not accurately reflect the longer-term secular rate of sea level rise because interannual and decadal variations in atmospheric pressure, water temperature, salinity, and other properties can cause sea level to fluctuate dramatically. Tide gauge data, on the other hand, record changes over a much longer period in the Adriatic. [9] Tide gauge data in the Adriatic have been analyzed previously by a number of authors [Emery et al., 1988; Douglas, 1991; Douglas, 1997; Tsimplis and Spencer, 1997; Orlic´ and Pasˇaric´, 2000; Woodworth, 2003; Marcos and Tsimplis, 2008]. Previous analyses of relative sea level rise recorded by Adriatic tide gauges generally provide rates between 0.4 and 1.4 mm/yr. The wide range of rates depends, in part, on the relative durations of the tide gauge records used for the study. Relative rates observed at Venezia, Italy, range from 1.3 to 7.3 mm/yr [Emery et al., 1988; Emery and Aubrey, 1991, Woodworth, 2003]. The

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Figure 3. Complex interplay between sea level rise, interseismic subsidence, and uplift/subsidence associated with GIA. wide range observed at Venezia is a consequence of variable amounts of anthropogenic influences, mainly pumping of groundwater, which has varied in intensity through time. Tide gauge rates observed at Venezia are an extreme case that we do not consider further. Not including Venezia, the average relative sea level rate inferred from Adriatic tide gauges during the 20th century has been estimated at 1.0 ± 0.2 mm/yr [Woodworth, 2003]. [10] Submerged coastal habitats could also potentially provide a record of relative sea level change averaged over the past 1000 – 2000 years, but such archeological data are scarce and poorly constrained in the Adriatic region. Flemming [1992] provides an uncertain average rate of relative Adria sea level rise of 2.3 ± 3.0 mm/yr for the last 2000 years. Another potential record of relative sea level change is provided by tidal notches found in the northern Adriatic [e.g., Benac et al., 2004]. The notches are located at depths ranging between 0.5 and 0.6 m below the sea level. However, the age of these notches is poorly known. If they are fairly recent, as suggested by similar notches found in Roman jetties [Fouache et al., 2000], then they may be consistent with sea level rise of 0.3 mm/yr. [11] Tide gauges record sea level relative to crustal rocks, and are thus affected by both absolute sea level variation and vertical crustal motions (Figure 3). The most ubiquitous and longest-wavelength vertical crustal motions affecting the Adriatic coast result from GIA associated with the loss of the Fennoscandian ice sheet. As discussed in section 1, existing GIA models for the Adriatic region are subject to large uncertainties and provide a range of different predictions, both in magnitude and sign. The applicability of GIA models in far-field regions, like the Adriatic, critically depends on the viscosity structure within the Earth, ice history, and model resolution and parametrization [e.g., Mitrovica and Davis, 1995], leading to large uncertainties in the predicted rates of vertical land motion. This uncertainty hampers our ability to infer secular (absolute) sea level rise from relative sea level data, but should not hinder analyses of relative vertical rates as long as the wavelength of the GIA signal is large relative to the tectonic signal and tide gauge network station spacing. [12] Short spatial wavelength variations in observed tide gauge trends could potentially signify vertical tectonic motions of the coastal rocks independent of the longwavelength trends associated with eustatic sea level change and GIA, particularly in the convergent margin setting.

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Along the eastern Adriatic coast, tectonic deformation is dominated by the collision of (at least) two subaqueous continental microplates with the Eurasia plate [Calais et al., 2002; Battaglia et al., 2004; D’Agostino et al., 2008]. These microplates (Figure 1), commonly referred to as the northern and southern Adria microplates, respectively, are the descendants of a larger late Paleozoic promontory of the Africa plate that collided with Eurasia during Cretaceous time, and subsequently, fragmented into independent microplates during Cenozoic time [Channell et al., 1979; Anderson and Jackson, 1987; Oldow et al., 2002; Battaglia et al., 2004; Rosenbaum et al., 2004]. Because these microplates’ kinematics are variable, their interaction with Eurasia along the eastern margin of Adria also varies. [13] Present-day counterclockwise rotation of these Adria microplates relative to Eurasia is apparent from earthquake focal mechanisms [Anderson, 1987; Anderson and Jackson, 1987] and GPS geodesy [Calais et al., 2002; Battaglia et al., 2004; Grenerczy et al., 2005; Serpelloni et al., 2005; Bennett et al., 2008; D’Agostino et al., 2008]. In northeast Italy, rotation of the northern Adria microplate results in north northeast directed convergence across the eastern Alps at a rate of 2 mm/yr [D’Agostino et al., 2005] (Figure 1). The locus of eastern Alps shortening is 90 km inboard from the coast. Strain accumulation associated with this convergence probably does not affect tide gauges located in Croatia. Along the northern Croatian coast, to the south and east of the Alps, the northeast motion of the northern Adria microplate is accommodated primarily by strike-slip faults mapped across the trunk of Istria peninsula (Figure 1). This transverse motion is not expected to cause significant vertical crustal motion, and should therefore not contribute appreciably to relative sea level rise recorded by tide gauges in northern Croatia. Northeast shortening along the southern Croatian coast (Figure 1) is the result of the counterclockwise rotation of the southern Adria microplate with respect to Eurasia. In contrast to the small amount of crustal motion expected for northern Adria, the elastic interseismic deformation model of Bennett et al. [2008] predicts subsidence along the southern coast of up to 0.6 mm/yr.

3. Observations [14] The data set that we consider consists of the tide gauge record of relative sea level along the eastern margin of Adria (Figure 2). Systematic measurements of sea level in the Adriatic began as early as 1859 in Trieste [Leder, 2002], administered by Academica Nautica and Osservatorio Marittimo in Italy. The Hydrographic Office of the AustroHungarian Navy also installed tide gauges at Rijeka, Losinj, Zadar, and Hvar around the same time for the purposes of scientific research and navigation safety. However, by the year 1918, these Austro-Hungarian tide gauges were decommissioned, with the exception of Trieste. The decommissioned tide gauges never resumed operation and the location of these early data is presently unknown. The earliest data available for Trieste are from 1875 [Raicich, 2007]. The rest of the tide gauges in operation today were installed as early as 1929. The locations of the tide gauges used in this study, as well as the tide gauges that we found to be not suitable, are shown in Figure 2. For our study, we concentrate on data from six tide gauges with a sufficient

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Table 1. Detailed Description of Tide Gauges Used in This Studya Site Triesteb Rovinj Bakar Split Harbour Split Rt Marjana Dubrovnik

Type A.OTT A.OTT A.OTT A.OTT A.OTT A.OTT

Kempten Kempten Kempten Kempten Kempten Kempten

Interval of Operation

Interval of Data Used

Authority

1859 – present 1955 – present 1929 – 1939 and 1949 – present 1929 – 1941 and 1947 – present 1952 – present 1954 – present

1955 – 2006 1955 – 2006 1955 – 2006 1955 – 2006 1955 – 2006 1955 – 2006

I.T.S. H.I.S. A.M.G.I.Z. H.I.S. I.O.F.S H.I.S

a From Leder [2002]. Authorities responsible for tide gauge maintenance: H.I.S. (Hydrographic Institute-Split), A.M.G.I.Z. (Andrija Mohorovicic Geophysical Institute Zagreb), I.O.F.S (Institute of Oceanography and Fisheries Split), and I.T.S. (Istituto Talassografico). b Station Trieste has been equipped with a digital recording system since 1987, sampling time 1 h, while still maintaining the graphic recorder.

number of observations to determine precise rates (Figure 2, Table 1). The characteristics of these gauges, including period of operation and instrument types, are listed in Table 1. [15] The data set used for this study was obtained from the Permanent Service for Mean Sea Level (PSMSL, http:// www.pol.ac.uk/psmsl). PSMSL distributes both the Revised Local Reference (RLR) data and metric data. The RLR data set has been reduced to a common datum and checked for inconsistencies prior to public availability, whereas the metric data set has not been corrected for datum shifts and has been entered in the databases directly as received from the authorities. Thus, we use the annual average RLR data set. [16] Stability of tide gauges maintained by the Hydrographic Institute of Croatia (HHI) has been regularly checked and confirmed by spirit leveling between benchmarks located on the tide gauge housing and other benchmarks in the vicinity that are assumed to be stable [Leder, 2002]. The tide gauge data set is homogenous in the sense that the same type of tide gauge instrument has been used at all stations in our study. The only appreciable data gaps are associated with the occurrence of World War II during the period of 1941– 1945 (Table 1). Uncertainties associated with the sea level measurements are not reported by the PSMSL; therefore, we assumed that all data points are of equal weight and that errors are adequately represented as independent samples from a zero-mean Gaussian distribution. Accordingly, we estimate the a posteriori variance of the distribution using the chi-square per degree of freedom statistic. [17] Correlation among the Adria tide gauge records is higher than 0.98 for all sites, suggesting that a large amount of the variation in the tide gauges is in common mode. In section 4, we develop an analysis technique by which any systematic error that is common to all sites is absorbed into parameters representing common-mode relative sea level variation, isolating the spatially variable signals associated with local secular sea level change relative to the crustal rocks. [18] The CGPS data set that we consider builds upon the work of Bennett and Hreinsdo´ttir [2007]. There are some important differences between our analysis and this previous analysis. We describe these differences, in turn, below. [19] We acquired and analyzed phase data from a much larger set of data than used by Bennett and Hreinsdo´ttir [2007], including 789 CGPS stations located throughout Europe and elsewhere around the world for the period of 1994 through the end of 2008. Our data set includes observations for stations AJAC, CAGL, ELBA, GENO, MARS, NICE, and SOPH (Figure 1), which were used by Bennett and Hreinsdo´ttir [2007] to define a central Mediterranean

fixed reference frame. Our data set also includes stations DUBR, SPLT, TRIE, and TRI2 located along the eastern side of the Adriatic Sea. Our analysis includes all available CGPS stations along the Adriatic coast that have been in operation for more than two years. Distances between the tide gauge and colocated CGPS station pairs Trieste – TRIE, Trieste– TRI2, Split Harbor– SPLT, and Dubrovnik–DUBR are 6.9, 2.1, 0.0, and 5.6 km, respectively. [20] Another important difference between our analysis and that of Bennett and Hreinsdo´ttir [2007] is that we analyzed the GPS phase data using an updated version of the GAMIT software package, version 10.3 [King and Bock, 2002]. The new software incorporates Institute of Geological Sciences, U. K. (IGS) absolute antenna phase center models for both satellite- and ground-based antennas, which greatly improves the accuracy of vertical rate estimates by mitigating reference frame scale and atmospheric mapping function errors [e.g., Schmid et al., 2005]. We applied a pole tide correction and ocean-loading model FES2004. We used the Niell mapping function (NMF) [Niell, 1996] for both hydrostatic and nonhydrostatic components of the tropospheric delay model. We used a priori orbits and Earth orientation parameters obtained from the International Global Navigation Satellite Systems (GNSS) Service (IGS), but we estimated adjustments to these a priori parameters. The phase data were batched by UTC day, and weighted according to an elevation angle dependent error model using an iterative analysis procedure, whereby the elevation dependence was determined from the observed scatter of phase residuals. We broke the data set into 50 station subnets and resolved integer phase ambiguities within each subnet. In addition to daily site position, we used GAMIT to estimate daily adjustments to satellite orbital parameters, Earth orientation parameters, and hourly piecewise linear zenith and horizontal gradient tropospheric delay parameters. We used the forward Kalman filter capability of GLOBK [Herring, 2002] to estimate site velocities from the daily position estimates. Site velocities were determined relative to the ITRF05 reference frame [Altamimi et al., 2007], utilizing the full variancecovariance matrix associated with the geodetic parameter estimates for each daily GAMIT subnet solution. We estimated phase center offsets to account for shifts associated with radome changes and coseismic displacements, but we relied on the phase center models when antennas were changed. We estimated two phase center offset parameters for site MARS, whereas Bennett and Hreinsdo´ttir [2007] estimated only one. Our GPS velocity estimates for AJAC, CAGL, ELBA, GENO, MARS, NICE, SOPH, DUBR, SPLT, TRIE, and TRI2 relative to the ITRF05 frame are listed in Table 2. The uncertainties for rates relative to the ITRF05

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Table 2. Vertical Crustal Motion of CGPS Stations Used in This Studya ITRF05

CMVF

Site

Rate

s

Rate

s

CAGLb GENOb NICEb SOPHb MARSb AJACb ELBAb TRIE TRI2 SPLT DUBR Meanc WRMSd

1.51 1.38 2.17 1.54 1.30 1.38 1.71 1.5 1.6 0.4 0.2 1.58 ± 0.02 0.2

0.03 0.06 0.07 0.07 0.10 0.05 0.07 0.1 0.2 0.3 0.1

0.03 0.16 0.56 0.00 0.20 0.16 0.17 0.1 0.1 1.1 1.7

0.13 0.26 0.30 0.30 0.43 0.21 0.30 0.4 0.9 1.3 0.4

a ITRF denotes rates relative to ITRF05, and CMVF denotes rates relative to the Central Mediterranean vertical reference frame (CMVF). All values are given in mm/yr. Uncertainties listed for ITRF05 are formal uncertainties, while uncertainties listed for CMVF have been scaled upward by factor of 4.3. b Sites used to define the CMVF. c Mean value of CMVF sites. d Scatter among CMVF sites.

frame, listed in Table 2, represent the formal uncertainties associated with the GLOBK velocity estimates. Before comparison of the GPS rates with tide gauge trends, we calculated realistic rescaled velocity uncertainties, as described in the following paragraph. [21] To facilitate interpretation of geodetic and tide gauge rate estimates, we defined a local Central Mediterranean fixed reference frame. Following Bennett and Hreinsdo´ttir [2007], we estimated a vertical translation to minimize the RMS rate of seven long running stations in the Central Mediterranean region (AJAC, CAGL, ELBA, GENO, MARS, NICE, and SOPH). Unlike Bennett and Hreinsdo´ttir [2007], we did not utilize uplift rate estimates derived from Holocene terrace data, as they do not reduce the RMS scatter of residual rates, possibly indicating that the GPS estimates have reached a level of precision that exceeds the precision of the estimates based on terrace data. The RMS scatter of the vertical velocities at these seven sites relative to the Central Mediterranean reference frame thus defined is 0.2 mm/yr. Vertical rates for the frame defining sites and Adria coastal sites relative to this reference frame are listed in Table 2. The formal uncertainties that arise from straightforward least squares analysis of GPS data typically underestimate the true errors due to temporal correlations in the phase and daily positioning residuals associated with imperfections in the models used to reduce the data [Blewitt and Lavalle´e, 2002; Bennett, 2008] and sources of stochastic noise [e.g., Williams et al., 2004]. To compensate for such correlations, velocity estimates may either be obtained using stochastic models representing correlated error processes [e.g., Williams et al., 2004; Langbein, 2008; Teferle et al., 2008], or uncertainties may be rescaled upward a posteriori by some amount after standard least squares error propagation [e.g., Davis et al., 2003]. One method for estimating realistic velocity uncertainties is to use the scatter among the frame defining sites as a measure of vertical rate precision [e.g., Davis et al., 2003]. This method provides a conservative measure of precision

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because it represents the sum of observation errors and unmodeled site motions. For our case, the c2 per degree of freedom of the frame defining sites’ vertical rates relative to the Central Mediterranean frame is 18.5, assuming zero real relative vertical motion among the sites. If our assumption of zero relative motion is correct, then the c2 statistic may be used to derive an a posteriori estimate of the data variance following traditional least squares error propagation. If there is some amount of real relative site motion, however, the c2 statistic would reflect variance associated with both observation errors and unmodeled site motion. In this latter case, the c2 statistic should provide an estimate of variance that exceeds the true observation error. Accordingly, we rescaled the velocity estimate uncertainties for the Central Mediterranean (AJAC, CAGL, ELBA, GENO, MARS, NICE, and SOPH) and Adriatic (DUBR, SPLT, TRIE, and TRI2) sites listed in Table 2 by a factor of 4.3, the square root of the c2 per degree of freedom. Because this scaling was derived using the velocity estimates themselves (as opposed to using time series residuals), it inherently accounts for velocity errors arising from time-correlated errors [e.g., Williams et al., 2004] and other sources of velocity estimation error such as limitations in the kinematic models [Blewitt and Lavalle´e, 2002; Bennett, 2008].

4. Tide Gauge and GPS Analyses [22] In this section, we develop analysis techniques for investigating the implications of the Adriatic Sea tide gauge record for sea level rise and vertical crustal motion along the eastern margin of Adria. Vertical crustal rates from tide gauges have been previously determined using a variety of methods. For example, the method proposed by Sjo¨berg [1987] uses tide gauge differences to infer relative secular land uplift rates. By forming differences, this method mitigates systematic effects common to both stations used. The limitation of this method is that it operates pairwise, and thus, does not fully exploit the information about common-mode variation afforded by a network of stations. Davis et al. [1999] developed a method to analyze tide gauge data based on a network of stations, explicitly including a term for interannual sea level variations, which was assumed to be a common mode across the network. As we show in section 4.2, the method proposed by Davis et al. [1999] is useful for studying secular relative sea level, but does not distinguish between common-mode and spatially variable components of the secular trend. In sections 4.1 and 4.2, we develop a new method that combines the strengths of these previous methods. Our new method yields rates that represent deviations from the mean motion, thus separating common mode from spatially variable secular trends. The motivation for developing our method is to isolate spatial variations in secular relative sea level trends from interannual variations, common-mode sea level change, or very long wavelength geodynamic motions of the crust. [23] One limitation of the current implementation of our method is that observations must exist for all stations at all epochs analyzed. Interpolation of time series to fill missing data points is complicated by very large interannual variability and would likely bias the results. However, for the current study, given the richness of the Croatian tide gauge record, this restriction has only a minor effect on our

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Figure 4. Tide gauge time series. Data have been offset along the y axis (which gives sea level in meters) for presentation purposes. analysis. We anticipate that an extended version of our analysis method may be developed to utilize all available data to provide improved, unbiased estimates in the future. [24] The first step in our analysis was to identify the subset of the epochs for which each station recorded data. Accordingly, we excluded all observations prior to year 1955. We did not use data after 2006 because, at the time of this study, data were unavailable through the PSMSL following year 2006. After reducing the data set, we found observations from 46 years and from six sites that met this criterion, as shown in Figure 4. Data from the Venezia site were not included in the study since, as discussed in section 2, irregular relative sea level rise observed at Venezia is not likely to represent a tectonic signal, nor common-mode signal associated with Adriatic sea level variations. [25] Following Davis et al. [1999], we consider the following model relating relative sea level records to spatially and temporally variable parameters:

is an N-dimensional vector containing the K tide gauge measurements at each of the P stations, m is an Mdimensional parameter vector representing the secular rates ri and constant offsets ai, and n is a K-dimensional vector representing all common-mode displacements bj. The N M matrix A relates m to d. Similarly, B is an N K matrix relating n to d. The variable e is an N-dimensional vector representing independent identically distributed measurement errors. [27] In general, the space spanned by the columns of the matrix A overlaps with the space spanned by the columns of the matrix B. The problem of determining m and n is underconstrained and solutions are nonunique. We address this nonuniqueness by hierarchical inversion. [28] Our objective is to study the pairs of vectors m and n that minimize the functional

Li tj ¼ ai þ ri tj þ bj þ eij ;

where k k2 represents the Euclidian norm. Due to the nonuniqueness of the problem, there are an infinite number of combinations of m and n that minimize the functional Y. However, not all of these solutions are equally meaningful. Below, we show two end member solutions, distinct in their way of treating the common-mode motion. The first method separates all common-mode variation from spatially varying signal. The second method, which is similar to the method developed by Davis et al. [1999], separates secular variations, including the common-mode component of secular variation, from spatially coherent nonsecular motion.

ð1Þ

where Li(tj) is the observed annual relative sea level average for the ith site at epoch tj, ai and ri represent temporally constant sea level offset and rate, respectively, specific to site i, bj represents a common-mode offset that is identical for all sites at epoch tj, and eij is an unknown observation error. We assume that the error processes can be approximated by white noise sequences from a zero-mean Gaussian distribution of unit variance. [26] Our objective is to solve for ai, ri, and bj simultaneously. For each of the P stations, there are K observations. Thus, we begin by writing equation (1) in matrix form d ¼ Am þ Bn þ e:

ð2Þ

Yðm; nÞ ¼ k d Am Bn k2 ;

ð3Þ

4.1. Method I [29] For this end member solution, we first consider the minimization problem

The vectors and matrices represented by this equation are described as follows: Let N = P K and M = 2P. Variable d 7 of 13

Yðm; ^ nÞ ¼ minK Yðm; nÞ n2