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Strategy for Future Measurements of Very-Low-Frequency Sea-Level Change 14 INTRODUCTION Tide-gauge records of long duration (on the order of 1 century) appear to be dominated by two processes: (1) noisy fluctuations with time scales on the order of 1 dec- ade and root-mean-square (rms) amplitudes of 50 mm, and (2) a long-term trend on the order of 1 mm/yr (Figure 14.1~. The decade fluctuations have horizontal coherence scales commensurate with the size of ocean basins (Figure 14.2), and they appear to be associated with what is loosely called El Nino. The long-term trend appears to be global and has been thought to be associated with climate changes of glacial periods. Table 14.1 summarizes some possible numbers, and Figure 14.3 sketches an associated spec- trum. We have taken a = 30 m for the rms amplitude of the ice-age-like fluctuations and typical period scales of 10,000 yr. This corresponds to a frequency of f = 10 ~ cycles per year (cpy) and an rms rate of 2 fa = 20 mm/yr, which is 20 times the present measured rate. It is hard to account for this 20:1 discrepancy. The geologic record of sea level and of glaciation prevents us from taking a much smaller amplitude than 30 m. Perhaps the current rate of rise is unusually small because we are approaching a climatic optimum; in fact, there is some evidence that the rate of sea-level rise was an order of magnitude larger around 10,000 yr ago. 221 WALTER MUNK, ROGER REVELLE, PETER WORCESTER, and MARK ZUMBERGE Scripps Institution of Oceanography DETECTABILITY OF LONG-TERM TREND There have been many discussions about the long-term 1-mm/yr sea-level rise. The question is asked whether this is now accelerating or decelerating. The following very simple calculation shows the difficulty in making such a prognosis. The difficulty has to do with the large ampli- tude of the decadal fluctuations. Suppose we form two averages, one over each half of the past century, and compare the two numbers for an indication of the long- term trend. For any single measurement of sea level, the high-frequency "noise" associated with the decadal fluc- tuations is 50 mm rms. The band width is such that we can form independent samples every 5 yr, or 10 samples in 50 yr. The rms error in a 50-yr average is then 50 mm/~101°5 = 15 mm. The rms error in the difference between the two 50-yr averages is 15 x (2~°.s = 21 mm. The expected difference in sea level associated with the long-term rise is 50 mm, about twice the noise level. This rough calcula- tion is in general accord with what is shown in Figure 14.1. There are a number of ways for improving the estimate of the long-term rise. The usual way is to form a global average. From Figure 14.2 we estimate that one can get 10 spatially independent samples, and in this way the decadal noise in the difference between the two 50-yr averages is reduced from 21 mm to 21/~101°5 = 7 mm, as compared to
222 can · 0 I ' ' i i 1 ~' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' 1 Period rms amplitude rms rate 1910 1920 1930 1940 1950 Current rate FIGURE 14.1 Honolulu mean sea level after removal of all tidal components and all noise above 2 cpy (after Munk and Cartwright, 1966~. the long-term trend of 50 mm. This gives a quite respect- able signal to noise ratio of 17 db, but the existence of pronounced local biases malces it difficult to realize a noes . . . . gain rom spaclal averaging. CRUSTAL NOISE The trouble with the above procedure is that the "solid" crust to which the tide gauges are anchored moves up and down at rates that are comparable to the rate of sea-level rise. Such being the case, the "crustal noise" rather than El 120° 150° E 180° W Boo - . N Do - n° - ... S - 30°; l5oo 1 20° 1 1 1 ~ , 1 1 _ DEC 1977 Y_ at_ . _ _ .. :::::::: :~::: :` , ~ .! _ Am_ 9 1 1 1 1 1 1 1 1 1 1 1 1 ,~ . ~ 150° E 180° W 150° 120° 90° FIGURE 14.2 Maps of sea-level anomaly (from Wyrtki and Nakahoro, 1984) for December 1975 and December 1977. Contours show sea-level anomalies in millimeters after removal of seasonal cycle. The two cases were selected for their great contrast. WALTER MUNK et al. TABLE 14.1 Periods, Amplitudes, and Rates of Sea-Level Fluctuations Climatologic al El Nino - like Fluctuations Fluctuations 1O,OOO yr 30 m 20 mm/yr 1 mm/yr 10 yr 50 mm 20 mm/yr Nino-like noise becomes the limiting factor in a global estimate. If the crustal movement could be independently measured, then the global estimates of sea-level rise could be vastly improved. THERMAL EXPANSION VERSUS CHANGES OF OCEAN MASS In considering fluctuations in sea level hats, we distin- guish between external processes involving variations in the total mass per unit area and those associated with internal changes in the density distribution (without much altering the total mass). The former processes are associ- ated with hydrostatic pressure fluctuations (pgh) on the seafloor; the latter, steric processes, give bottom pressure fluctuations that are smaller by 2 or 3 orders of magnitude or are absent altogether. A warming by 1°C of the entire water column would raise sea level by about 0.2 m. A warming by 10°C would One lo4 m2 too Cpy 1 10-; 1 0 I ce Age -l ike fluctuation - _ 10 ~10-4 10-3 10 ' cycles per year El Ntno -l ike fluctuation \ ~ I 1 1 1 1 10-1 1 FIGURE 14.3 A cartoon of the sea-level spectrum. The climatological (ice-age-like) oscillations are taken to have an rms amplitude of 30 m and a time scale of 10,000 yr. The El Nino-like events have an rms amplitude of 50 mm and a time scale of 10 yr. The inferred rms rate of sea-level change is 20 mm/yr for the climatological oscillations (20 times the present rate) and also 20 mm/yr for the El Nino-like fluctuations. There is no firm evidence for the spectral gap at 10-2 cpy, or for the drop-off below 10-4 cpy; the ice-age-like and E1 Nino-like fluctuations may be plateaus on a monotonic red spectrum.
STRATEGY FOR FUTURE MEASUREMENTS OF VERY-LOW-FREQUENCY SEA-LEVEL CHANGE raise sea level by about 8 m (much more than 10 x 0.2 m because the coefficient of thermal expansion increases with temperature). Implication is that a 30-m variation in sea level cannot be attributed to thermal expansion. The conclusion is that the rise over the last few thousands of years must have been largely due to external processes, mainly owing to the decay of continental ice sheets. The situation is not clear with regard to the rise by approximately 1 mm/yr during this century. The glacio- logical evidence (Meter, 1984) seems to rule out a contri- bution by much more than 0.3 mm/yr from mountain and alpine glaciers; an 8 percent decrease in the current glacial volume would correspond to a 50-mm rise in global sea level. A 50-mm rise could also be brought about by a 0.3 percent decrease in the volume of the Antarctic ice sheet (corresponding to a lowering of the ice-sheet surface by 6 m). Most Antarctic glaciologists, however (Robin, 1985; National Research Council, 1985), do not believe that such a change is taking place. At the same time, there is evidence of a warming by 0.4°C of global surface temperature during the twentieth century (Gornitz et al., 1982), and this yields a steric rise of perhaps 0.6 mm/yr when interpreted in terms of a model of vertical ocean diffusivity. But the evidence is not convincing, and the model is demonstrably poor. In summary, the ice-age-like fluctuations are probably largely external, but the associated long-term trend over the last 50 yr may include an important steric component. Now with regard to the decadal fluctuations, Roem- mich and Wunsch (1984) demonstrated that they are largely steric. Independent measurements could be used to sub- tract the steric (El Nino-like) noise from the sea-level measurements and so obtain a better estimate of the exter- nal component in the long-term trend. MEASUREMENT OF g The position of the sea surface relative to a benchmark on land is affected both by sea-level change and by verti- cal motions in the crust beneath the benchmark. Tide- gauge measurements alone may be insufficient to monitor true changes in sea level the heights of the tide gauges themselves must be monitored with respect to a worldwide reference frame. One technique that can be brought to bear on the prob- lem is the absolute measurement of g, the acceleration due to the Earth's gravity. As an observer moves away from the center of the Earth, g decreases at the fractional rate of 3 parts in 10~°/mm. The local value of g can currently be determined by portable instruments (Zumberge et al., 1982) to within an uncertainty of about 1 part in 108, providing a sensitivity to height changes on the order of 30 mm. Thus, periodically making accurate determinations of g near tide 223 gauge installations can provide a cost-effective means of separating sea-level change from tectonic motion. The nature of the method used to make absolute gravity measurements makes them well suited to a search for very-low-frequency signals such as those related to sea- level change. In a modern absolute gravity meter, a mass is made to fall freely in a vacuum while its position as a function of time is determined by a laser interferometer. By calculating the acceleration of the falling mass, g is determined in terms of the wavelength of a stabilized laser and the frequency of an atomic time standard. Both are referenced to absolute standards having stabilities surpass- ing 1 part in 109. Other factors limit the accuracy obtained to about 1 part in 108, but we believe that this can be improved by a factor of 2 or 3 in the next decade (Faller et al., 1983~. The important point is that g is determined absolutely, and thus the data are not likely to become contaminated by instrumental drift. Incorporating gravity data into an analysis of a tide- gauge record can be a complicated problem. A rise in sea level will affect gravity because of the additional water mass and through deformation of the crust by the added load. Conversely, a local expansion of the underlying crust will move the observation point and displace sea water, both of which affect g. An order-of-magnitude assessment of the contribution to change in g from the two mechanisms sea-level change as opposed to tectonic motion relies on two simple numbers. The first (mentioned above) is the free air gradi- ent of gravity. The fractional change in g for a height displacement z is ~/g =3 x 10 /mm. (14.1) This will be roughly the size of the gravity signal accom- panying a vertical displacement caused by crustal defor- mation. The other value we need is the gravitational attraction of a laterally infinite layer of water having thickness t. Relative to the nominal value of g (about 9.8 m/s2), the attraction is ~g/g =4 x 10 /mm. (14.2) If we ignore the crustal loading effect (its sign and magni- tude are comparable to the infinite layer effect) (War- burton and Goodkind, 1977; Goad, 1980), we see that the change in gravity accompanying an apparent change in sea level due to crustal deformation is an order of magnitude larger than the gravity change caused by a real variation in sea level. Thus repeated gravity measurements can be useful in distinguishing one mechanism from the other. Other geodetic methods can be used to monitor the vertical positions of tide gauges, and any strategy de
224 signed to address the sea-level problem should take advan- tage of them. For example, electromagnetic distance measurements to satellites using radio waves or lasers can determine relative geodetic heights with an uncertainty ranging from 20 to 50 mm between stations separated by 100 km or more (Christodoulidis and Smith, 1983; Strange, 1984~. Over intercontinental distances, very-long-baseline interferometry can determine heights to 100 mm or better (Shapiro, 1978; Herring, 1984~. As these methods im- prove, they are bound to exert a profound influence on the strategy for assessing global sea level. PRESSURE ON THE SEA FLOOR The best long-term measurements of seafloor pressure made to date have relied on a careful characterization of the drift characteristics of the pressure sensor from labora- tory measurements so that the pressure record could be corrected for sensor drift after completion of the experi- ment. Wearn (Wearn and Baker, 1980; Wearn and Larson, 1982), for example, believed that he was able to correct data obtained using Paroscientific quartz-crystal pressure sensors with sufficient precision to reduce the residual drift to between 1 and 3 mbar (approximately 10 to 30 mm) per year at an ambient pressure of about 500 dbar (approximately 500 m). This corresponds to a relative accuracy of 0.002 to 0.006 percent, or 2 x 10-5 to 6 x 10-s. To determine long-term trends in sea level, one would like to be able to make measurements with substantially greater precision; an increase of 1 mm in 4000 m of water, for example, represents a change of 2.5 x 1O-7. One does not want to make measurements at much shallower depths because the pressure change then has both barotropic and baroclinic components. In order to measure such trends, one therefore needs to either (1) devise a pressure sensor with lower drift or (2) calibrate the sensor against a pri- mary standard in situ. A pressure sensor with significantly lower drift than the Paroscientific quartz sensor used by Wearn does not exist; a transducer with lower drift would therefore have to employ an entirely new technology that is unknown at this time. In the remainder of this section we will therefore discuss the precision that is achievable with in situ calibrations. One could imagine configuring the system in two dis- tinct ways. 1. A valve could be used to switch the input port of the pressure sensor between ambient ocean pressure and a reference pressure used to calibrate the sensor. This tech- nique requires a sensor with adequate short-term stability, which may be a problem. The reference pressure would need to be close to ambient pressure in order to minimize hysteresis effects in the sensor. WALTER MUNK et al. 2. A reference pressure could be used with a differen- tial pressure sensor of much lower full-scale pressure than required to directly measure ambient ocean pressure. This configuration takes advantage of the fact that drift is typi- cally proportional to the full-scale pressure of the sensor, whereas the absolute size of the fluctuating signal is the same for either absolute or differential sensors. In either case, the best candidate for providing the refer- ence pressure is a high-quality, oil-operated, piston-gauge deadweight tester. In such a unit a calculable pressure is generated in a fluid by placing a known mass on a piston of known diameter. The absolute accuracy achieved with a high-quality piston gauge is 0.01 percent, or 1 x 10~. But it is important to remember that what is required is relative, not absolute pressure changes. The important parameter is the reproducibility or stability of the pressure generated by a piston gauge. The National Bureau of Standards (now National Institute for Standards and Tech- nology NIST) routinely finds short-term repeatabilities at the 1 ppm (1 x 10-6) level when intercomparing high- quality piston gauges, but long-term (years) differences in some cases have significantly exceeded this level (D. R. Johnson and C. Tilford, NIST, personal communication). NIST believes that 10 ppm is a more realistic expectation for long-term stability; they are not able to verify the absolute stability at any better level since 10 ppm is the irreproducibility of the dimensional measurements of the pistons and cylinders. This is a rather disappointing state of affairs. A relative stability of 1 x 10-5 corresponds to 40 mm out of 4000 m. This is a factor of 2 to 6 better than Wearn claims to have achieved by measuring and removing the drift of his pres- sure sensors, but is substantially less stable than desired. It is conceivable that one might achieve better than 10 ppm stability in the deep-sea environment; the temperature is relatively stable (fluctuations are measured in millidegrees centigrade in the deep Pacific), the instrument would not be handled, and the piston weights would never be changed. Testing this speculation would require a substantial re- search project in which a number of piston gauges would be maintained in a simulated deep-sea environment for periods in excess of a year. Such a test would reveal noise that is uncorrelated between units. Systematic errors can be determined only by comparison to a better standard; no such standard exists. The best that one would be able to do is compare piston gauges from several manufacturers. An alternative approach would be to deploy the sensor at significantly shallower depths, decreasing the fractional accuracy required. At shallower depths one might also expect the drift problem to be more tractable, since one would be dealing with lower pressures. The disadvantage would be that the pressure signal would then have a sig
STRATEGY FOR FUTURE MEASUREMENTS OF VERY-LOW-FREQUENCY SEA-LEVEL CHANGE nificant baroclinic component. However, even at 100 m, corresponding to 1-mm precision for a stability of 1 x 10-5, one would expect the baroclinic component to be substan- tially attenuated relative to that measured by a surface tide gauge (since the largest ocean temperature changes occur in the upper ocean). If lower drift could be achieved at pressures corresponding to a few hundred meters depth, one might be able to move the sensor somewhat deeper. But this is clearly not the most desirable approach. If one were to pursue the use of piston gauges on the seafloor for long time periods, a number of practical prob- lems would need to be addressed. Some of the less obvi- ous ones include the following: 1. Although NIST is of the opinion that a high-quality, oil-operated piston gauge would function continuously for a year, there is no evidence of anyone that has done it for longer than two weeks (D. R. Johnson and C. Tilford, NIST, personal communication). 2. The axis of the gauge must be aligned to the vertical to better than 1 milliradian. 3. The gauge would have to be installed on the seafloor in such a way that it would settle by less than 1 mm/yr. 4. The external pressure applied to the piston by the gas in the pressure vessel used to house it must either be measured or eliminated by housing it in a vacuum. 5. The temperature of the piston and cylinder must be monitored to allow compensation for the approximately 9 ppm/°C temperature coefficient of the effective area. Many other detailed considerations would be required for making high-precision measurements; the ones given above are only examples. Success with this approach would prompt the desire to construct instruments to be deployed for multiyear periods, with real time data readouts on shore. Fiber-optic technology currently under active de- velopment should make it economically feasible to con- nect the instruments via cables to shore-based data record ers. INVERTED FATHOMETER Consider a fathometer looking upwards from the seafloor. The one-way acoustic travel time is Sh, where S = 1/C is the sound slowness and h is the total water thick- ness. In a homogenous ocean, the departure in travel time iS lit = S(Z - z ~ as a result of departures z and z surface c ' surface c in the surface and crustal elevations, respectively. We define z - t/S SO that z = z - z . IF IF surface c For an external contribution to the water budget OF = z - z . The effect of steric disturbances is twofold: it leads b c to an additional component, Is, in the surface elevation, and it changes the sound speed in the water column. In considering the effect of a long baroclinic wave in a 225 two-layer ocean, the upper ocean is characterized by a thickness hi, density pi, temperature 0~, and salinity so, with a corresponding h2, P2, 02, and s2 for the lower ocean. With a change in layer thickness from h. to (h. + ah. ), j = 1,2, but leaving Hi and si unchanged, the condition of constant mass per unit area requires that piths + p2bh2 = 0, or Gh2 = -(P1/P2)6h1 (14.3) The steric change in sea levels equals z = Gh + Oh = 2 1 ah . s 1 2 P2 ~ This can be written The change in acoustic travel time (one-way) is fit = Sldhi + S26h2 - SOZF- (14.5) St = S2 dh lo- 5 A. (14.6) The first term is S2ZS' which is very nearly the increase in travel time associated with the steric rise in sea level. Typical values are (P2 - P1~/P2 = 0.001 and (S2 - S14/S2 = 0.01 so that the second term dominates. In terms of the temperature differential be, we have bp/p =-a~p and bS/ S = -ocCp, with a _ 0.13 x 10-3/°C and a = 3 x 10-3/°C. Thus the ratio of the second to the first term in brackets is p = a/a _ 23. (14.7) The result is that the second (negative) term dominates. A thickening of the upper layer (positive Chic raises the sea level by zs = t(P2 - p~/P23Gh~ and thus increases the one- way travel time by Szs; but the relative thickening of the warm (high speed) upper layer decreases travel time, and the latter effect dominates. We can write the result of a purely steric disturbance in the form ZIF Zs(l 2), p >> 1. (14.8) TOMOGRAPHY The inverted fathometer method is associated with a steep acoustic path and by its nature depends on a combi- nation of surface elevation and the variable interior field of sound speed. Tomography depends on near-horizontal refracted paths that do not intersect the surface and so can be used to estimate Is directly. An advantage of the method is that it provides a spatial average (order 1000 km). This reduces the mesoscale noise (100 km, month scale) in the estimate of the decadal fluctuations (104 km, decade scale). Munk and Wunsch (1985) estimated an rms
226 noise level of 5 mm in the determination of the mean annual steric sea level, using tomographic transmissions. INTERPRETATION What could be learned when some of these measure- ments are combined with the tidal recording? Let Zc designate the crustal height relative to some suitable reference; changes in Zc can be estimated from the gravity measurements. Let Is be the steric sea level; an increase in Is can arise from thermal expansion and haline contraction. The for- mer effect is dominant. (The annual tide, with a typical amplitude of 10 cm, is largely the result of such tempera- ture changes.) An important consideration is that the coefficient of thermal expansion increases appreciably with increasing temperature. Accordingly, most of the changes in Is are associated with the wand ocean water above the thermocline, both because the shallow temperature changes are relatively large and because the coefficient of thermal expansion of the upper ocean waters is relatively large. A distinction can be made between the effects of a net change in heat content (e.g., due to radiation, evaporation) and the local change in temperature associated with flow diver- gence, as in baroclinic waves. Let Zb designate the barotropic sea level. Here a dis- tinction can be made between the effects of a global change in the ocean's mass arising from glacial melting, and the local changes associated with flow divergence. An ex- ample of the latter is the buildup of the gyre center at the expense of the flanks, with the net gyre mass remaining unchanged. We consider the records obtained from a tide gauge, a bottom-pressure recorder, an inverted fathometer, and a grav~meter: ZTG Zb s c' ZBP b c' ZIF - Zb + Z (1 - p) -Z. ZGR c Without bottom pressures, which are difficult to measure, there are three equations in the three unknowns Zb' Zs' and Zc With bottom pressure, the problem is overdetermined. The system of equations is oversimplified. As an ex- ample, a rise in global sea level due to global melting of ice will lead to an increase in (due to the added water mass beneath the gravimeter) and a further increase due to the compression of the crust from loading, in addition to any effect from tectonic crustal uplift. The system of equations is intended as an illustration of how a particular set of observations can be used. There are many other ways too. Repeated hydrographic surveys yield a direct measure of the steric level (the classical WALTER MUNK et al. dynamic height). Changes in the crustal elevation can also be surmised by the perturbation of satellite orbits. A PROPOSED STRATEGY We propose that two existing tide stations be instru- mented to include a gravimeter, a bottom-pressure gauge, and an inverted fathometer. There is much to be said for combining this new instru- mentation with a program proposed by NOAA (1985) as a contribution to the World Ocean Circulation Experiment (WOCE). This program consists of three principal com- ponents: (1) new tide gauges that record digitally and have a high degree of linearity, (2) very-long-baseline interfer- ometry (VLBI), and (3) the Global Positioning System (GPS). VLBI and GPS provide a highly accurate (+1 cm) global reference frame with a role similar to that of the gravimeter. We propose Hawaii and Bermuda as the two sites for developing the advanced concepts of sea-level measure- ments. Both stations are relatively free of meteorological noise (Brown et al., 1975), and have long-time series of tidal measurements. Hawaii is a site for the proposed NOAA sea-level program. Bermuda has been the site of the Panulirus measurements of steric sea level since 1954 (see Roemmich, Chapter 13, this volume). As a second phase of the proposed program of en- hanced sea-level measurements, one should establish a limited global network using the experience gained at the prototype stations (Hawaii and Bermuda). If local adjust- ment for crustal movement and steric sea-level fluctua- tions is successful, then presumably the steric component of the long-term trend can be estimated from measure- ments at a relatively small number of stations. The steric component (if any) of the long-term trend will be difficult (14 9) to extract from the 50-mm rms background noise of the E1 Nino-like fluctuations. For 10 spatially independent sta tions, each forming 5 independent samples (this takes 25 yr for a 5-yr decorrelation time), we thus have 50 degrees of freedom in the global mean, or +7 mm standard devia tion. During this 25-yr period we can expect a change of 10 to 25 mm in steric level resulting from the long-tenn trend. The problem is made even more difficult by the expectation that the long-term trend itself will change during this 25-yr interval. REFERENCES Brown, W., W. Munk, F. Snodgrass, H. Mofjeld, and B. Zetler (19751. MODE bottom experiment, J. Phys. Oceanogr. 5~1), 75-85. Christodoulidis, D. C., and D. E. Smith (1983~. The Role of Satellite Laser Ranging through the 1990's, NASA/Goddard
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