Interpreting Sea Level Records

The seeming inability of investigators during the last few decades to arrive at a consensus concerning the precise rate of global sea level rise, or even how to approach the problem, has led some authors to conclude that global sea level rise cannot be measured at all. Barnett (1984) states that ". . . it is not possible to uniquely determine either a global rate of change of sea level or even the average rate of change associated with the existing inadequate data set." Emery and Aubrey (1991) state that (p. 176) "At present, we cannot discover a statistically reliable rate for eustatic rise of sea level alone . . . ." Pirazzoli (1993) is the most pessimistic, declaring ". . . the determination of a single sea-level curve of global applicability is an illusory task." Douglas (1995) and Gornitz (1995b) provided detailed analyses and interpretation of estimates published since 1980, and concluded that the situation is not so bleak. In his determinations of global sea level rise and (lack of) acceleration, Douglas (1991,1992,1997) stressed the importance of using very long records. He concluded that the large differences in estimates of sea level rise could be explained in nearly all cases by the selection criteria used by investigators, especially the use of short (<60 years) records and data from sites affected by plate tectonics. Gornitz (1995b) further pointed out that the use of long records and explicit model corrections for glacial isostatic adjustment (GIA) in most cases leads to values nearer to 2 than to 1 mm/yr. But controversy continues. In this section we examine carefully some individual tide gauge records that show explicitly how the characteristics of sea level change can radically influence the results for the computed sea level trend at a site. In particular, the role of record length will be shown to be of the utmost importance in determining trends of relative sea level.

To measure water level appears to be simple, but a time series of sea levels measured by a tide gauge contains much information. As noted, the gauge is designed to "filter out" some information, that is, changes of water level occurring over periods of a few seconds due to waves; these changes are better measured by other instruments. Tide gauges respond well to the familiar daily and semi-daily tides, whose amplitude is typically of the order of a meter. This is 500 or more times the amount of the yearly rise of sea level at most places, so removal of tidal signals from sea level data by a low-pass filter is necessary for investigations of long-term sea level change. This can be done very precisely if desired, since the motions of the sun and moon and rotation of the earth determine tidal periods (Pugh, 1987). Simpler digital filtering techniques also suffice. Some other fluctuations of coastal sea level, such as shallow water effects of tides, wind-driven setup, storm surges, or precipitation runoff, are not so easily modeled. However, averaging the water level over monthly or annual intervals effectively eliminates these small or limited duration events. Other larger and temporally more enduring water level variations, such as the seasonal-interannual fluctuations due to El Nino, create significant difficulties in determining a trend of sea level.

As an example of the signal content of a sea level record, consider the monthly mean water levels in Fig. 3.2. These data from the tide gauge site at the Scripps Institution of Oceanography in La Jolla, California, were obtained from the PSMSL Web site. The PSMSL provides monthly mean data in columnar format, making it very easy to analyze the data with a spreadsheet program. The Scripps gauge, operated and maintained as a part of the National Water Level Network by the National Ocean Service (NOS) of the National Oceanic and Atmospheric Administration (NOAA), is an RLR site that has been in operation since 1925. It is located at the end of a long research pier facing the open ocean, and provides an excellent record of sea levels.

The data shown in Fig. 3.2 are relative sea levels (RSLs). This merely means that the origin is arbitrary and that changes in water level cannot be distinguished from vertical movements of the land or platform on which the

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Figure 3.2 Monthly mean sea level at La Jolla, California.

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Figure 3.2 Monthly mean sea level at La Jolla, California.

gauge is located. An absolute system of sea levels, one in which all tide gauges are positioned in a common coordinate system, is highly desirable. If the coordinates of a tide gauge could be independently monitored for change in an absolute global geodetic reference frame, the observed trend of sea level at a location could be freed of contamination from vertical land movement. To this end, there is an international effort under way to provide such a system at the millimeter level of precision using a wide variety of modern geodetic technologies. For more information, consult the Harvard—Smithsonian Center for Astrophysics Sea Level Page at http://rgalp6.harvard.edu/index_

rsl.html, the International GPS Service for Geodynamics Web site at http:// igscb.jpl.nasa.gov/, and the International Earth Rotation Service at http:// hpiers.obspm.fr/. As discussed in Chapter 6, the results are promising. Within a decade it appears that vertical crustal movements at enough tide gauge sites will be known accurately enough in an absolute geodetic reference frame to solve the vexing problem of local and regional elevation changes of the land.

The monthly mean sea levels in Fig. 3.2 display an upward trend of a few millimeters per year, with an additional "noise" of about 100 mm. The real nature of the high-frequency component of the record is obscured because of the scale of the graph. Figure 3.3 shows the data for only the decade 1980-1990, along with a 3-month running mean curve. What may have looked like random noise in Fig. 3.2 is revealed in Fig. 3.3 to be very systematic in character. The obvious annual component of the signal arises (at least at this particular location) largely from thermal expansion and contraction of the

Figure 3.3 Monthly mean sea level at La Jolla 1980-1990.

upper layer of the ocean; the remaining components are small and irregular and probably meteorological in origin. Also easily visible as a superimposed multiyear anomalous period of elevated sea level is the effect of the 1982-1983 El Niño event. El Niño is primarily a warming phenomenon of the tropical Pacific Ocean, but also produces changes of sea level that propagate north and south along the west coast of North and South America (Chelton and Davis, 1982) raising sea level well beyond the norm. Higher water level during El Niño can have important consequences. Large waves generated by Pacific storms struck the California coast during times of unusually high sea level in the winter of 1982-1983 and caused severe beach and bluff erosion in some areas.

For an interesting example of decadal and longer variations of sea level, consider Fig. 3.4. It displays annual mean sea levels for Annapolis and Baltimore, Maryland, and Atlantic City, New Jersey, since 1930. The sea levels are in units of millimeters, and only relative changes can be inferred from each series. In addition, the series are offset by an arbitrary amount from one another for the sake of clarity and each has an added running 3-year mean through the data. The high correlation of the series at long periods is very obvious, and gives confidence that the multiyear fluctuations are real. (In Chapter 7 Sturges and Hong provide the explanation for them; their origin is meteorological.) Multiple sea level records from nearby tide gauges are useful for verifying that details of a record of sea levels are real, and not a result of instrument error.

Figure 3.4 U.S. mid-Atlantic annual mean relative sea levels.

What may be surprising about the series in Fig. 3.4 is that an interannual correlation of sea levels actually exists in these records. Atlantic City is on the open coast, and Annapolis and Baltimore are located in different branches of a very large estuary (the Chesapeake Bay) that receives heavy seasonal inflows of fresh water from rivers. But viewed from the perspective of interannual and longer periods of water level variation, Chesapeake Bay water level is in dynamic equilibrium with the nearby open ocean. There are a number of other gauges in the Bay, and they all display this same correlation at interannual and longer periods. Douglas (1991,1992,1997) evaluated the tide gauge records in his determinations of global sea level rise and acceleration in this manner and placed them into morphological groups based on their apparent correlation at low frequencies with their neighbors. This ensured that no oceanographically coherent region received undue weight in the global average of regional sea level trends simply because it happened to have a large number of tide gauge sites.

Sometimes relatively nearby gauge site records disagree in their interannual variations. When this occurs, there is always the possibility that one of the records is not suitable for some reason. A good example of this problem is given by a pair of long Australian sea level records. In his investigation of global sea level rise, Douglas (1991) found that the long-period variations of relative sea levels at Sydney and Newcastle, only a few 100 km apart, were 180° out of phase for nearly half of their common data interval. The trends derived from these records also differed by a factor of two. These records were not used in his analysis, nor in his reanalysis (Douglas, 1997) because no explanation could be found for the difference.

Another pair of nearby tide gauges, Buenos Aires and Quequen in Argentina, also give sharply different trends (1.5 vs 0.8 mm/yr) over their rather long (85 and 65 years, respectively) records. In addition, the records of the two sites do not in any way resemble each other morphologically even though the gauges are only about 400 km apart. Fortunately, a plausible explanation for the difference can be found in this case. It is a surprising one and offers an excellent illustration of the problems involved in obtaining a reliable trend from a series of water levels recorded by a tide gauge.

Figure 3.5 displays annual values of mean sea level at Buenos Aires, with the final 7 years (1981-1987) shown as open squares; these data after 1980 were not used to determine the linear regression trend result of 1.1 mm/yr shown in Fig. 3.5. It is remarkable that eliminating only a small part of the record (i.e., the 7 years after 1980) decreased the estimated trend of water level from 1.5 to 1.1 mm/yr, much closer to the Quequen value of 0.8 mm/yr. What caused the very rapid rise of water level after 1980? Figure 3.6 provides the probable answer. This figure displays detrended and normalized values of smoothed monthly means of the Buenos Aires sea level record and the negative of the Southern Oscillation Index (SOI). The SOI, which is computed from the difference of barometric pressures at Tahiti and Darwin, is negative

1900 1910 1920 1930 1940 1950 1960 1970 1980 Figure 3.5 Annual mean relative sea level at Buenos Aires.

during warm El Nino-Southern Oscillation (ENSO) events and positive during cold ones. See Diaz and Markgraf (1992) and Trenberth (1997) for more detailed discussions of the ENSO phenomenon. The CD-ROM accompanying this book contains a sequence of global monthly mean sea surface temperature

Buenos Aires RSL Negative of the SOI

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 Figure 3.6 Buenos Aires relative sea level and the negative of the SOI.

anomaly maps from 1982 to 1999 prepared by Dr. Charles Sun of the National Oceanic and Atmospheric Administration (NOAA). These maps are shown in rapid succession and present a "movie" of the phenomenon that clearly illustrates the global extent of it. The most up-to-date version of the animation can be found at www.noaa.nodc.gov.

SOI values used to prepare Fig. 3.6 were obtained from the NOAA Climate Prediction Center (CPC) Web site http://www.cpc.ncep.noaa.gov. The signs have been reversed on the SOI values in Fig. 3.6 in order to make clearer the correlation of the SOI and sea level. There are some gaps in the SOI record, but Fig. 3.6 shows that there is an obvious correlation (r = 0.46) of sea level at Buenos Aires with the SOI. Some of the largest ENSO events are especially well reflected in sea level fluctuations there. Figure 3.7, the equivalent of Fig. 3.6 for Quequen relative sea levels, in contrast shows no significant correlation (r = 0.03) with the SOI. How is this disagreement between Buenos Aires and Quequen sea level records and the SOI possible? The answer almost certainly lies in the geographic situation of these two sites. Quequen is south of Buenos Aires on the open South Atlantic Ocean. No correlation with ENSO events is seen there, nor expected; for one to occur in analogy to the effect of the ENSO on western North and South American water levels, a coastal wave would have to propagate around the southern tip of South America and up the east coast, a physically unrealistic scenario. Buenos Aires has a different situational aspect. It is located well inside an estuary that drains a very large portion of northern Argentina, southern Brazil, Paraguay, and Uruguay. Increased rainfall during ENSO years would have the effect of decreasing the

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Figure 3.7 Buenos Aires relative sea level and the negative of the SOI.

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Figure 3.7 Buenos Aires relative sea level and the negative of the SOI.

salinity of the water in the estuary and increasing its flow, both leading to an increase of water level. So the correlation can plausibly be attributed to an indirect ENSO effect, rather than a direct one. But the impact on water levels at Buenos Aires is nearly as dramatic as seen on U.S. Pacific Coast water levels.

The example of Buenos Aires and Quequen demonstrates that the usual approach to determining a trend of sea level, ordinary linear regression, has significant limitations. Specifically, the linear regression model assumes that the signal or time series consists of a trend with added Gaussian (i.e., normally distributed) random noise. To the extent that a series of water levels deviates from this model, the computed trend and its uncertainty will be corrupted. Thus the linear regression model is at best an approximation to the real nature of a series of water levels.

From a statistical point of view, there are two main issues in using linear regression to determine sea level trends from tide gauge records. First, it is apparent from the examples in this chapter that there is serial correlation of the measurement residuals about the trend line. This demonstrates that the measurements contain information that cannot be accommodated by the model, so that the effective number of degrees of freedom is less than the number of observations minus two. Second, the character of the residuals is inconsistent. They seem to lack the character called stationarity, that is, their statistical properties at least appear to be nonuniform over the series. (Wunsch (1999) has pointed out very clearly how such appearances can be deceptive.) In these finite length water level records, the lack of uniformity of the signal character over the record creates great difficulties in identifying and determining an underlying trend.

To illustrate the problems of model inadequacy, it is instructive to analyze the Buenos Aires sea level trend in terms of formal uncertainties, and compare to actual results. The standard deviation, cr, of the trend obtained from a linear regression is calculated from (see any elementary statistics text)

where MSE is the sum of squares of the residuals about the trendline divided by the number of observations less 2, and the summation is over all values of the times Application of this formula for the standard deviation of the trend at Buenos Aires gives a = 0.22 mm/yr for both 1905-1980 and 1905-1987. Fortuitously, the shorter span of data from 1905-1980 gives the same uncertainty (to two significant figures) of the trend as the longer span. This occurs because even though shorter, the MSE is smaller due to elimination of the large residuals after 1980. But the actual results for the trend upon extending the record from 75 years (1905-1980) to 83 years (1905-1987), only 9%, increases the apparent trend of sea level by about 36% (1.1 to 1.5 mm/yr). Obviously, the formal uncertainty of the trends obtained from Eq. (1) is too small. The explanation for this anomaly lies in the nature of the sea level signal. We have noted that the residuals about the trend line are in fact serially correlated; that is, adjacent residuals have a tendency to be similar in size. In addition, the anomalous increase of sea levels after 1980 is inconsistent with the assumption of stationarity, and has an extreme impact on the trend estimate.

A quantitative measure of the serial correlation of the residuals about the trend line for Buenos Aires sea levels shown in Fig. 3.5 is presented in Fig. 3.8, where members of adjacent residual pairs (i.e., lag 1) are plotted against each other. If there were no serial correlation, there would be no trend; the existence of the trend shows that there is a general tendency for adjacent residuals to increase or decrease together. For a discussion of serial correlation in residuals, see Draper and Smith (1981).

Maul and Martin (1993) used the lag 1 correlation of residuals to analyze their long series of annual mean sea levels at Key West, Florida, and found that the effective sample size was less than one-half the number of observations in that case. The results shown here for Buenos Aires are similar. The lag 1 serial correlation indicates an increase of the formal error by about 40%, that is, to about 0.3 mm/yr, which is significant, but not enough to explain the extreme sensitivity of results to inclusion or exclusion of 7 years of data after 1980. That sensitivity mostly exists because of the anomalous increase of sea level near the end of the record associated with the extreme 1982-1983 ENSO event.

In regard to stationarity of the residuals, the assumption of ordinary linear regression is violated by the unpredictable large anomalies caused by ENSO events (or anything else). The anomaly of sea levels at the end of the Buenos Aires record has an especially large effect on the apparent overall trend of sea level because of its location in the series. If an anomaly is near the average of the independent variable, there is little effect on the derived trend, but if

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Figure 3.8 Lag 1 autocorrelation of residuals for Buenos Aires.

it is near either end of the series, the effect can be very great. Should the data after 1980 be discarded? In this case, we can identify the reason for the anomaly (i.e., a large ENSO event) and use that knowledge to justify elimination of the anomalous data. The more reasonable estimate of the underlying long-term trend of sea level at Buenos Aires is the one lacking data after 1980, that is 1.1 rather than 1.5 mm/yr.

The effect of El Niño on the Buenos Aires record was unexpected because of the location of Buenos Aires. For a case where El Niño is known to cause readily observable changes of water level, consider the San Francisco sea level variations in Fig. 3.9. This figure, prepared in the same manner as Fig. 3.6, shows smoothed, detrended, and normalized values of monthly mean values of relative sea level (RSL) at San Francisco and the negative of the SOI as far back (to 1882) as available from the CPC web site. The correlation of San Francisco sea level anomalies with the SOI is significantly more striking than in the case of Buenos Aires. This is especially so after 1935. Trenberth (1997) has stated that SOI values prior to 1935 are reconstructions in many cases from barometric pressures other than from Tahiti and Darwin, and so may be a less accurate representation of ENSO. To the extent that U.S. West Coast sea level fluctuations reflect ENSO events, the high correlation (r = 0.63) of the SOI with San Francisco water level anomalies after 1935 bears out this contention. The correlation for the SOI and sea level anomalies prior to 1935 is r = 0.43.

The examples of San Francisco and Buenos Aires give clear warning that interannual and longer variations of sea level can have a very large effect on

- - San Francisco RSL - Negative of the SOI

- - San Francisco RSL - Negative of the SOI

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Figure 3.9 San Francisco monthly mean relative sea levels and the negative of the SOI.

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Figure 3.9 San Francisco monthly mean relative sea levels and the negative of the SOI.

trends computed even from long series. Figure 3.10 offers another view of the severity of this problem by displaying the trend of sea level at San Francisco for a 50-year sliding window. Each point in Fig. 3.10 is the trend of sea level in millimeters per year computed for the entire period of the preceding and following 25 years. Obviously an arbitrarily selected 50-year trend of sea level at San Francisco, which even changes sign at several points, does not reflect the underlying trend there! This example of dependence of sea level trend on epoch and record length, like the one at Buenos Aires, is larger than most, but not remarkably so.

What is revealed by these examples is that every sea level record must be examined, and a decision made as to whether the record is long enough to determine the underlying long-term trend of sea level. Some authors have used sea level records as short as 10 or 20 years in their analyses, which is in no case adequate. The most basic reason for the disagreements of authors concerning the rate of global sea level rise for the 20th century has come from a lack of a consensus on what constitutes an adequate length of sea level record. Increasingly, however, longer records are being used. Peltier and Tushingham (1989, 1991) in their analyses concluded that records of 50 years and longer gave an acceptable trade-off between adequate length and number of available records. Douglas (1991,1992) argued that at least 60-year records were needed. Later Douglas (1997) found even better (i.e., more precise) results for global sea level rise by using records longer than 70 years. The case of Buenos Aires discussed here shows that even 80 years may not be long enough in some special cases unless identifiable extreme anomalies in

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Figure 3.10 Fifty-year trends of sea level at San Francisco.

the sea level record can be identified and eliminated from the determination of the trend.

For a broader look at the role of record length in computed sea level trends, consider Fig. 3.11. It shows 20th-century trends of sea level obtained by linear regression for all tide gauge records from PSMSL RLR sites, plotted against record length. These trends have been corrected for GIA using the ICE-3G values of Peltier and Tushingham (1989). Vertical movement of the land due to GIA is a component of the sea level trend at any location. Unless a correction is applied, sea level trends determined from individual tide gauge records cannot be meaningfully compared, or aggregated to form an estimate of a global trend. Note that the longest records have GIA-corrected trends that converge to a positive value well in excess of 1 mm/yr with increasing record length. Figure 3.11 also shows clearly that estimates of sea level trends made from records as short as 10 or 20 years (e.g., Nakiboglu and Lambeck, 1991; Barnett, 1984; Emery and Aubrey, 1991) can be badly biased by the interannual-to-decadal variability water level.

There is some scatter visible in Fig. 3.11 even for trends determined from the longest record lengths. Most of this disparity arises from vertical crustal movements other than GIA and from data problems. The ubiquity of these issues concerning tide gauge records is illustrated by the redetermination of global sea level rise by Douglas (1997). There are about 175 RLR records >50 years in length but only 24 of these were used in this redetermination of global sea level rise; the others were rejected for geophysical and other reasons.

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