• Ashkenazi, V., R. Bingley, G. Whitemore, and T. Baker, 1993: Monitoring changes in mean sea level to millimeters using GPS. Geophys. Res. Lett.,20, 1951–1954.

    • Crossref
    • Export Citation
  • Barlow, R., 1989: Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences. John Wiley and Sons, 204 pp.

  • Beers, Y., 1962: Introduction to the Theory of Error. Addison–Wesley, 66 pp.

  • Carter, W., Ed., 1994: Report of the Surrey workshop of the IAPSO tide gauge bench mark fixing committee. NOAA Tech. Rep. NOSOES0006, 81 pp. [Available from National Ocean Service, 1305 East-West Highway, Silver Spring, MD 20910.].

  • Cheney, R., B. Douglas, and L. Miller, 1989: Evaluation of Geosat altimeter data with application to tropical Pacific sea level variability. J. Geophys. Res.,94, 4737–4748.

    • Crossref
    • Export Citation
  • ——, L. Miller, R. Agreen, N. Doyle, and J. Lillibridge, 1994: TOPEX/Poseidon: The 2-cm solution. J. Geophys. Res.,99, 24555–24563.

    • Crossref
    • Export Citation
  • Christensen, E., and Coauthors, 1994: Calibration of TOPEX/Poseidon at Platform Harvest. J. Geophys. Res.,99, 24465–24486.

    • Crossref
    • Export Citation
  • Dillon, W., and M. Goldstein, 1984: Multivariate Analysis: Methods and Applications. John Wiley and Sons, 587 pp.

  • Douglas, B., 1991: Global sea level rise. J. Geophys. Res.,96, 6981–6992.

    • Crossref
    • Export Citation
  • Hayne, G., S. Hancock III, and C. Purdy, 1994: TOPEX altimeter range stability estimates from calibration mode data. TOPEX/Pos. Res. News,3, 18–20.

  • Minster, J.-F., C. Brossier, and P. Rogel, 1995: Variation of the mean sea level from TOPEX/Poseidon data. J. Geophys. Res.,100, 25153–25161.

    • Crossref
    • Export Citation
  • Mitchum, G., 1994: Comparison of TOPEX sea surface heights and tide gauge sea levels. J. Geophys. Res.,99, 24541–24553.

    • Crossref
    • Export Citation
  • Mitrovica, J., J. Davis, and I. Shapiro, 1994: A spectral formalism for computing three-dimensional deformations due to surface loads. Part II: Present-day glacial isostatic adjustment. J. Geophys. Res.,99, 7075–7101.

  • Morris, C., and S. Gill, 1994: Evaluation of the TOPEX/Poseidon altimeter system over the Great Lakes. J. Geophys. Res.,99, 24527–24540.

    • Crossref
    • Export Citation
  • Murphy, C., P. Moore, and S. Ehlers, 1996a: Preliminary study of the use of tide gauge data to unify altimetry data from separate satellites. Adv. Space Res., in press.

  • ——, ——, and P. Woodworth, 1996b: Short-arc calibration of the TOPEX/Poseidon and ERS-1 altimeters utilizing in situ data. J. Geophys. Res.,101, 14191–14200.

    • Crossref
    • Export Citation
  • Nerem, S., 1995: Global mean sea level variations from TOPEX/Poseidon altimeter data. Science,268, 708–710.

    • Crossref
    • Export Citation
  • Rapp, R., Y. Yi, and Y. Wang, 1994: Mean sea surface and geoid gradient comparisons with TOPEX altimeter data. J. Geophys. Res.,99, 24657–24667.

    • Crossref
    • Export Citation
  • Sciremammano, F., Jr., 1979: A suggestion for the presentation of correlations and their significance levels. J. Phys. Oceanogr.,9, 1273–1276.

    • Crossref
    • Export Citation
  • Wyrtki, K., and T. Bongers, 1987: Sea level at Tahiti—A minimum of variability. J. Phys. Oceanogr.,17, 164–168.

    • Crossref
    • Export Citation
  • ——, and G. Mitchum, 1990: Interannual differences of Geosat altimeter heights and sea level: The importance of a datum. J. Geophys. Res.,95, 2969–2975.

    • Crossref
    • Export Citation
  • View in gallery

    Illustration of how the TOPEX and tide gauge time series were paired. The open circle is the location of one of the tide gauges (at Papeete in French Polynesia) used in this study. The small dots mark the locations where TOPEX sea surface height time series are available. The larger solid dots mark the four TOPEX time series used to form difference series with the tide gauge sea level record. The TOPEX points lie at the same latitude rather than at the point of closest approach, but that is not critical to the results. Points at the same latitude are chosen because the meridional length scales of sea level variability are significantly smaller than the zonal scales for tropical stations, which form a large fraction of the available tide gauge locations.

  • View in gallery

    Locations of tide gauges used in this study. The locations of the 53 tide gauges remaining after the quality control checks described in the text are completed and are marked by solid circles. Note that each tide gauge can produce up to four difference time series (see Fig. 1), so there are fewer stations shown than the number of difference series (149) used in the final set of analyses.

  • View in gallery

    Histogram of the sample correlations (rmn). Estimates were computed for every possible pairing of the 149 difference time series. Typically there are order 60 degrees of freedom, and the standard deviation of the correlation estimate assuming zero true correlation is approximately 0.13 (see text).

  • View in gallery

    The fitted correlation function for the case of zero spatial separation. This component of the correlation function applies to difference time series derived from the same tide gauge but different TOPEX passes. The solid curve is the fitted function given by (9a) in the text. The solid dots and errors bars are bin-averaged correlation estimates derived from the data. The bins are chosen to have equal numbers of points in each. Note that the fitting was done with the actual data values, and the bin averaging is done only to aid the presentation of the result.

  • View in gallery

    As in Fig. 4 but for the case of small temporal separations that arise when the altimeter passes over two different tide gauges on the same orbit.

  • View in gallery

    Result of the initial analysis. The solid circles and error bars are the result of the TOPEX drift analysis using the tide gauge data. At this point the TOPEX data have not been corrected for the recently discovered algorithm error (shown as the solid line), which was in fact not known at the time the tide gauge analysis was done. The means have been removed from both time series because the tide gauge mean is arbitrary, but otherwise, no fitting has been done.

  • View in gallery

    Final result of the tide gauge determination of the TOPEX stability. The solid dots and error bars show the results of the tide gauge estimates of the TOPEX drift, as in Fig. 6, but the TOPEX data have now been corrected for the algorithm error, and the tide gauge analysis has been recomputed.

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Monitoring the Stability of Satellite Altimeters with Tide Gauges

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  • 1 Department of Marine Science, University of South Florida, St. Petersburg, Florida
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Abstract

A method is described for using tide gauge sea levels to monitor time-dependent drift in satellite altimetric measurements of sea surface height. The method depends on a careful assessment of the quality of the tide gauge measurements available for this application and also takes into account the degree of independence between the altimeter minus tide gauge differences in order to construct an optimal drift estimate and an accurate error estimate for it. The method is applied to the TOPEX altimeter measurements, and a recently discovered algorithm error, which resulted in a slow drift in the TOPEX sea surface heights, is exploited to evaluate the success of the tide gauge drift estimation. It is important to note that the tide gauge analysis was done without any prior knowledge of this error. The result is that the tide gauge analysis reproduces the drift due to the algorithm error to within 6 mm rms, which is comparable to the 5–6-mm internal estimate of the uncertainty of the drift analysis. The analysis is then made using the TOPEX data that have been corrected for the algorithm error and shows that the satellite heights are stable to better than 10 mm over the nearly 4 years of data available, although a drift on the order of 2 mm yr−1 remains, the source of which is unknown. This inferred stability is more than adequate for the majority of applications, although questions still exist for more demanding applications, such as the calculation of global sea level change. Limitations of the tide gauge analysis are discussed, along with potential improvements that might be possible in the future.

Corresponding author address: Dr. Gary T. Mitchum, Department of Marine Science, University of South Florida, 140 Seventh Ave. S., St. Petersburg, FL 33701.

Email: mitchum@marine.usf.edu

Abstract

A method is described for using tide gauge sea levels to monitor time-dependent drift in satellite altimetric measurements of sea surface height. The method depends on a careful assessment of the quality of the tide gauge measurements available for this application and also takes into account the degree of independence between the altimeter minus tide gauge differences in order to construct an optimal drift estimate and an accurate error estimate for it. The method is applied to the TOPEX altimeter measurements, and a recently discovered algorithm error, which resulted in a slow drift in the TOPEX sea surface heights, is exploited to evaluate the success of the tide gauge drift estimation. It is important to note that the tide gauge analysis was done without any prior knowledge of this error. The result is that the tide gauge analysis reproduces the drift due to the algorithm error to within 6 mm rms, which is comparable to the 5–6-mm internal estimate of the uncertainty of the drift analysis. The analysis is then made using the TOPEX data that have been corrected for the algorithm error and shows that the satellite heights are stable to better than 10 mm over the nearly 4 years of data available, although a drift on the order of 2 mm yr−1 remains, the source of which is unknown. This inferred stability is more than adequate for the majority of applications, although questions still exist for more demanding applications, such as the calculation of global sea level change. Limitations of the tide gauge analysis are discussed, along with potential improvements that might be possible in the future.

Corresponding author address: Dr. Gary T. Mitchum, Department of Marine Science, University of South Florida, 140 Seventh Ave. S., St. Petersburg, FL 33701.

Email: mitchum@marine.usf.edu

1. Introduction

We present a method for using sea level data from tide gauges to monitor low-frequency drift in sea surface height data from satellite altimeters, which can alternatively be viewed as providing a way to verify the long-term stability of the altimetric data. The application of the technique described here is to the data from the TOPEX instrument, which was launched in 1992 and continues to provide data as of this writing. The technique is, however, perfectly general and can be applied to any altimetric dataset. In the case of TOPEX, where the sea surface height time series are now nearly 4 years long and promise to continue for some time to come, and for any other present and future altimeter that will provide long time series, the monitoring of the long-term stability is important, as researchers apply these data to the study of interannual variations. And such stability is essential for analyses aimed at studying decadal variability, or long-term sea level change, as has been recently reported, for example, by Nerem (1995).

This need to evaluate the stability of time series is, of course, not particular to satellite altimeters. Any observation depending on mechanical devices cannot be assumed to be free of drift. For example, the tide gauges that are used in this study to evaluate the drift of the altimeters are certainly not assumed to be immune to such problems. Tide gauge sea levels are routinely, and continuously, monitored for drift by comparison to readings taken from tide staffs, which are sometimes also called tide poles. This is simply a calibrated rod placed near the tide gauge and read usually several times a week by a human observer. These measurements are noisier than the tide gauge measurements but are an admirably direct reading of the height of the sea surface. Differences between the tide staff readings and the contemporaneous tide gauge sea level data are monitored for low-frequency drifts. If any such drift is detected, then the tide gauge installation is inspected to determine the cause. The staff readings can also be used, given a sufficiently long averaging time, to calibrate the tide gauge sea levels in order to ensure a consistent zero point, or datum, as the instruments age or are replaced with different types. The ability of the historical tide gauge sea levels to determine global sea level trends to a precision of a few tenths of a millimeter per year (Douglas 1991) attests to the success of this strategy.

As compared to altimeters, the tide gauge measurements can also be considered a relatively direct, though spatially sparse, measurement of the sea surface height. The analogy to the tide gauge–tide staff system, in which the temporally sparse tide staff data are the more direct observations, is obvious. It could be argued, without being facetious, that ultimately the technique described in this paper amounts to using the tide gauges as an intermediary that allows the stability of the altimeters to be monitored by a large number of human tide observers reading sea level heights from their tide poles. Given the success of this approach in maintaining the quality of the global tide gauge network, this should be an encouraging thought.

In earlier studies that use tide gauge sea levels as ground truth for Geosat data (e.g., Cheney et al. 1989;Wyrtki and Mitchum 1990) or TOPEX/Poseidon data (e.g., Mitchum 1994; Cheney et al. 1994), the emphasis was on evaluating the success of the altimeter in capturing the ocean signals, usually by computing temporal correlations and standard deviations of the mismatch between the tide gauge sea levels and nearby altimetric sea surface heights. In the present case, the emphasis is on a careful analysis of the differences between the two. The ocean signals are assumed to largely cancel in such differences, allowing us to more easily search for systematic errors. In particular, these differences are examined for low-frequency trends that are taken to indicate altimeter drift. Obviously, the tide gauges are taken to be stable, but this assumption will also be examined in this paper.

The approach described in the paper is certainly not the only one possible. Other proposals include the use of satellite to satellite comparisons rather than satellite to in situ, for example, but the focus here is strictly on the latter approach using in situ data. To our knowledge, all the present methods for monitoring the altimetric heights make use of sea level data from tide gauges, although other types of in situ data, such as dynamic topography or surface velocity, could also be used. The approaches that use tide gauge data are all ultimately based on monitoring time series of differences between sea levels at one or more tide gauges and the corresponding sea surface heights observations from the altimeter in question. Of course, the differences between the tide gauge sea levels and the sea surface heights from the altimeter are not due solely to drift in the altimeter but also reflect the noise in each instrument. This can be actual instrument noise, or it can be due to real ocean signals present in one dataset and not the other. For example, at tide gauges along continental margins, signals due to coastally trapped waves or boundary currents will be observed but may not be captured by the altimetric heights that are measured farther from the coast. But given time series of the altimeter–tide gauge differences, an average can be taken at each point in time to allow the noise to cancel and the drift to dominate. The variance of the drift estimate will be proportional to the variance of the individual series that are combined and inversely proportional to the number of independent time series available.

These very basic ideas provide a simple way of characterizing the various approaches to the problem of deriving the most precise altimeter drift estimates. In the case of the TOPEX altimeter, one extreme is the approach taken at the project’s calibration site on the Harvest oil platform off California (Christensen et al. 1994). In this case only a single time series is used, but the excellent suite of ancillary measurements available at the Harvest site is quite effective at reducing the variance of the TOPEX minus tide gauge difference series well below what can be done at a typical tide gauge site. It is also important to note that because of the additional geodetic information collected at this site, the differences computed at the Harvest platform are absolute. This is a unique advantage of this approach, although we also note that promising results from absolute sea levels obtained with a tide gauge near a satellite laser ranging station in Europe have recently been reported (Murphy et al. 1996b).

An alternative approach has been described by Morris and Gill (1994). These authors note that tide gauge sea levels in lakes, the Great Lakes in North America in their study, should agree more closely with the altimeter heights simply because of the smaller signal levels in the lakes, as compared to the open ocean. Thus, the variances of the individual difference time series are relatively small compared to open ocean comparisons, although not as small as that obtained at the Harvest site. There are, however, multiple tide gauges available in the lakes, which can be averaged to increase the number of degrees of freedom and decrease the variance of the drift estimate in that fashion. It is not clear, however, how many of the lake tide gauges can safely be considered independent, and this approach would seem to benefit from using data from as many lakes as possible.

At the other extreme from the Harvest approach is the approach taken here. In this case, it is accepted that, as compared to Harvest, there will be relatively large differences between the altimetric sea surface heights and the tide gauge sea levels at any given point in space. The variance of the drift estimate is minimized by using a large set of tide gauges in order to maximize the number of independent series. This approach was suggested by Wyrtki and Mitchum (1990) in an analysis of the Geosat data and was improved upon and applied to the early TOPEX data by Mitchum (1994). The technique presented here is a significant improvement on the earlier method in that the variance of the drift estimate is much smaller.

The paper is organized as follows. In the next section the data processing is described, and the details of how tide gauges were selected for use in this analysis are given. In the following two sections the technique is derived. The paper concludes with presentation and discussion of the results obtained from applying the technique to the TOPEX data. This section includes an evaluation of the technique that was made possible by a recently discovered algorithm error in the TOPEX data. This error produced an artificial trend in the TOPEX heights and, since the time history of this trend is now known, it allows a check of how well the tide gauge analysis, which was done without prior knowledge of it, did in identifying the altimeter drift. Finally, a discussion of the limitations of this method, and possibilities for improving it in the future, is given.

2. Data processing

The altimeter chosen for the initial application of the technique is the TOPEX instrument, which is the United States component of the joint TOPEX/Poseidon project with France. This mission was launched in late 1992 and continues to provide data as of this writing. The TOPEX ground track is repeated every 9.9 days, which is taken as the temporal spacing for the estimation of the time series of the TOPEX minus tide gauge difference series. Data were used from cycles 6 to 129, which span almost 1230 days, or just over 40 months. Data from the first five cycles were not used due to uncertainties about the quality of the TOPEX data during this initial phase of the mission.

The TOPEX data processing was generally according to the recommendations of the TOPEX project, as outlined in the user’s handbook available with the data. Some differences, however, should be noted. First, we have used the improved orbits (JGM-3) and tide model (CSR 3.0) available from the Center for Space Research at the University of Texas. Second, we have not applied an inverted barometer correction because atmospheric pressure observations were not available at all the tide gauges and because the comparison of the actual sea surface heights from the altimeter and the sea levels from the tide gauges is considered to be more direct than a comparison of the pressure deviation at the sea surface inferred from an assumed hydrostatic balance via the inverted barometer adjustment. Third, we have included an adjustment for an internal calibration available from the project (Hayne et al. 1994). This calibration was made part of the recommended TOPEX data processing during the course of this work, and our data are consistent with that recommendation.

It should be carefully noted that while this paper was in preparation, and after the calculations had been completed, it was reported that there was an algorithm error in the processing of the TOPEX data before it was provided to users. This error is such that it produces a slow temporal drift in the TOPEX heights. This error has not been corrected in the initial analysis presented below. The reason for this is simple. This type of error is precisely the sort of problem that this analysis was designed to detect and therefore provides an excellent way of checking the usefulness of the tide gauge analysis given that the true drift caused by the algorithm error is now known. Basically, there was a drift in the data that the tide gauges provided an estimate for without any a priori knowledge. Then the time series of the actual TOPEX drift was specified, at least for any part of it due to the algorithm error, forming a valuable verification for this analysis. In the final section of the paper, a revised drift estimate, which was derived after correcting for the algorithm error, will be presented and discussed.

Turning to the processing of the tide gauge data, sea level data from 101 tide gauges were obtained from the WOCE “Fast Delivery” Sea Level Center operated as part of the University of Hawaii Sea Level Center. These data are available via anonymous ftp and are generally available for transfer within 1–3 months of data collection. This timeliness is important for this application, which seeks to provide a timely monitoring of the stability of altimeters that provide data within that same time frame. At each of the 101 stations, four difference series were obtained with the nearest TOPEX passes, two ascending and two descending, that bracket the tide gauge location (Fig. 1). The differences were formed by taking the TOPEX point at the same latitude as the tide gauge and subtracting the analogous daily sea level value, which has been low-pass filtered to eliminate diurnal and semidiurnal tidal signals. The tides have also been removed from the TOPEX data, but by using a tide model. In principle, the TOPEX data could be averaged for some distance along the ground track, or the TOPEX data could be taken from the point of closest approach to the tide gauge, but the differences are not very sensitive to such changes (Mitchum 1994).

Only time series with at least 50 valid differences (of 124 possible) were retained. In some cases the tide gauge data are gappy, but in most cases where less than 50 points are available it is because no data exists for TOPEX. Generally this is because the nearest pass is over land. There were 220 time series from 81 stations that had at least 50 points in the time series. Of these 220 only 8 had fewer than half of the 124 possible points available, and the median number available was 92. For each of these time series, the mean difference was removed because the tide gauge zero point is arbitrary, as discussed earlier. Also, 59- and 62-day harmonics were fit and removed to reduce noise due to tide model errors. These harmonics correspond to the alias periods in the TOPEX data for the S2 and M2 tidal constituents, respectively. These candidate series were then used to compute the standard deviation of the differences and the correlation between the TOPEX sea surface height and tide gauge sea level series used to form the differences. As discussed in the previous section, the best TOPEX, tide gauge pairs for the estimation of the drift are those that agree closely, which allows the oceanic height signals to cancel in the difference, leaving a series dominated by the drift. Series where the two did not agree well, which was defined to be where the standard deviation exceeded 100 mm or where the correlation was less than 0.3, were rejected. These choices are somewhat arbitrary, but using other values, or not rejecting points at all, does not change the drift estimates significantly. Rejecting these series leaves 176 time series at 68 sites.

The remaining difference series were next examined for large trends that might indicate land motion problems at the tide gauges. Note that a large trend in the tide gauge sea level was not considered cause to reject a station if the associated TOPEX data revealed a similar trend. This was the case, for example, at Honiara in the Solomon Islands and at Rabaul in Papua, New Guinea. Four stations did fail to meet this criterion: Socorro Island, Cabo San Lucas, Lobos, and Talara, all of which are located on the Pacific coasts of Central and South America. Rejecting the difference time series from these four stations leaves 168 series from 64 sites.

Finally, one additional set of stations were rejected after it was noted that a number of the difference series had relatively large seasonal variations. This is undesirable not only because it indicates that the altimeter and tide gauge are measuring different signals but also because the serial correlation associated with such low-frequency variations in the difference time series reduces the number of degrees of freedom in the series and makes it less useful. It was suspected that these seasonal signals could be due to coastal processes, such as boundary currents, coastally trapped waves, or freshwater input from large rivers. Such a tide gauge would not then be representative of the open ocean sea surface heights measured by TOPEX, and those time series should be rejected. To test for this effect, an annual harmonic was fit to each difference time series, and series where the annual amplitude was both greater than 2 cm and significantly different than zero were flagged. We decided to be conservative about rejecting gauges, however, and only sites that had all of the available series flagged (recall that each tide gauge is associated with up to 4 difference series) were rejected. In all there were a total of 11 stations (19 difference series) rejected by this criterion. All of these stations were along continental margins, which supports the interpretation in terms of coastal variability that is not captured by the altimeter. Of these 11 stations, 7 were from the west coast of North America, 1 was from the east coast, 1 was from Japan, and 2 were from Africa.

The net result of applying these various quality control checks to the tide gauge data is a set of 149 difference series from 53 tide gauge sites. The locations of these tide gauges are shown in Fig. 2. Finally, we point out that the final drift estimates are not highly sensitive to the particular subset of gauges used. The major effect of restricting the stations as described above is to reduce the variance of the time series of the drift estimates.

3. Description of the method

We start with a set of measurements of TOPEX sea surface heights minus tide gauge sea levels as a function of time t and at a number of tide gauge locations, which are denoted by the subscript n. These difference time series are defined as δnt, and the problem is then to use the δnt to produce an optimal estimate of the drift time series, which will be defined as Δt.

It should be reiterated that the differences are not absolute because the tide gauge sea levels are referred to an arbitrary zero point. Thus, it is only possible to monitor the time-dependent drift of the altimeter, and it is not possible to determine a constant offset. As stated above, however, the complementary analysis at the Harvest platfom can determine such offsets. If the tide gauges were to be placed in an absolute coordinate system, by the use of GPS receivers, for example, this could change. Also, it is considered important that the technique operate independently at each time step. This eliminates the need for any assumptions about the temporal form that the altimeter drift might take. For example, there is no assumption that the drift is linear in time. Since we wish to determine the drift estimate independently at each time step, the time subscript will be dropped for convenience. Thus, we want to find an optimal value of Δ given a set of measurements, δn, of it.

In discussing the method it is useful to consider the various contributions to the δn. Define the TOPEX height at a given time and at a particular tide gauge site to be
hnsnεn
where hn is the observed height, sn is the ocean signal, Δ is the altimeter drift, and εn is the sum of all of the other altimeter errors. Similarly, define the tide gauge sea level as
ηnsnlnε"n
where ηn is the sea level observation, sn is again the ocean signal common to both observations, ln is the error due to local effects near the tide gauge, and ε"n is the sum of all the tide gauge errors. With these definitions
δnhnηnεnlnε"n

From (3) it is seen that variance of the TOPEX minus tide gauge differences will be smaller at some locations than others. For example, some tide gauges may be more contaminated by local effects (ln) not observed by the altimeter. An example of this, signals due to coastally trapped variability along continental margins, was mentioned earlier and has also been discussed by Mitchum (1994). Even at island stations, the variances of the difference series can be quite variable. For example, at Papeete in French Polynesia the variance is relatively small due to generally low sea surface height variability in the region (Wyrtki and Bongers 1987), while the nearby station at Rarotonga has a larger variance due to propagating signals that are measured differently by the tide gauges and by TOPEX (Mitchum 1994). Therefore, it is necessary to determine a separate variance for each δn, which will be defined as σ2n.

The differing variances associated with the δn could be easily accommodated by defining Δ as a weighted average (e.g., Beers 1962), with the weights inversely proportional to the variances, σ2n, but this still assumes that the δn are independent. This assumption is not considered to be a good one. First, at least part of the difference between the altimeter and the tide gauge measurements is from real ocean signals in one instrument that are not observed in the other. There is no reason to assume a priori that these signals will be uncorrelated from one spatial location to another. Second, even if the differences are all due to noise, it still does not imply that the δn are necessarily independent. For example, if the TOPEX instrument noise has even a short temporal correlation associated with it, then this could cause correlations between the δn that are associated with different, and well-separated, tide gauges but which are relatively close together in time when measured along the altimeter ground track. Also, since the TOPEX heights are corrected for a variety of enviromental variables, which are not perfectly known, correlations in the errors in these corrections could be problematic. So simply downweighting points with larger variances could be counterproductive if it effectively restricts the data locations to a small geographical region and discards useful degrees of freedom in space. It is necessary to balance two possibly conflicting goals. That is, locations with large variance should be downweighted, but locations that are largely independent of the others should be retained even if the variances are somewhat larger.

For the moment, assume that at a particular time there are N estimates of the altimeter minus tide gauge differences (δn). Further, assume that the theoretical variances of the δn (σ2n) are known, as is the theoretical correlation function between the N estimates. Specifically, define ρmn as the correlation between δm and δn. A covariance matrix for the errors can then be formed with elements
Rmnσmσnρmn
We now rewrite (3) as
δnεn
where εn is simply the sum of all the error sources. The matrix R, with elements given by (4), describes the variances and covariances of the εn. Equation (5) is written in vector form as
δε,
where X is simply a column vector of length N with all elements equal to one. This problem is solved using weighted least squares (e.g., Dillon and Goldstein 1984), and the result is
ΔXTR− 1X−1XTR− 1δ
and the variance of Δ, defined as σ2, is given by
σ2XTR−1X−1
Several points are worth noting. First, if the δn are independent and have equal variances (say, s2), then it is easy to show that the drift estimate computed from (7) is given by a simple arithmetic average of the δn, and its variance is s2/N, as expected. Second, if the δn are independent, but their variances are not equal, then (7) takes the form of a weighted average, with the weights inversely proportional to the σ2n, which is also a well-known result (Beers 1962). Finally, we note that the formulation can be extended to more complicated, spatially varying, models of the drift simply by adding additional basis functions to the X matrix and making Δ a vector of coefficients. As will be discussed near the end of the paper, this may prove necessary in the future.

To summarize this section, given a vector of TOPEX minus tide gauge differences (δ) at any given time, and given a specification of the covariance matrix for the errors in these observations (R), Eqs. (7) and (8) allow us to compute the drift estimate (Δ) and its variance (σ2) at that time. This procedure is then repeated at each time step to determine the drift time series and the associated values of its variance. The variances of the δn and the correlation function were estimated from the observed δnt time series, as described in the following section.

4. Estimation of variances and correlation function

For each of the 149 difference time series selected in section 2 the sample variance (s2n) was computed, as were the sample correlations (rmn) with all the other difference series. Two things must be noted. First, in the derivation given above, the σ2n and ρmn are the theoretical variances and correlations for a single point in the time series. So using the time series estimates of the variances and correlations, s2n and rmn, requires assuming that the statistics are stationary, or that σ2n and ρmn do not change with time. Second, the σ2n and ρmn values should be computed from the noise in the time series; that is, the residual time series after the signal, which is the altimeter drift series to be determined, is removed. It was found, however, that simply using the raw difference series gives nearly identical results to an iterated approach that computed an initial drift estimate and then recomputed the residual statistics, etc. This is expected if the noise variance is much larger than the signal (drift) variance, which is in fact the case.

The noise variances σ2n were estimated directly from the s2n values. This is a reasonable estimate because the time series exhibit relatively little serial correlation, as evaluated using a runs test (Barlow 1989), and typically have about 75 degrees of freedom. This implies that the variance estimate is determined to about 16% (Barlow 1989). Estimating the true correlation function, ρmn, from the time series estimates of the correlation, rmn, however, is not so straightforward. In addition to the rmn values, the number of degrees of freedom associated with each rmn (Sciremammano 1979) was also computed. There were typically about 60 degrees of freedom, and the standard deviations of the rmn, assuming that the true correlation is zero, is thus on the order of 0.13. At lowest order, the distribution of the rmn values (Fig. 3) is quite consistent with a zero mean distribution with that variance, but closer inspection reveals a number of primarily positive outliers, indicating that some of the correlation is not consistent with an assumption that ρmn is zero for m not equal to n. But this correlation is superimposed on a larger noise background. So, in the case of ρmn, as opposed to σ2n, the estimate must be estimated from a set of noisy samples and should not be computed directly from the difference time series.

The values of the sample correlations, rmn, were extensively examined as a function of the distance separating the tide gauges used to form the difference series, which can be zero for difference series from the same tide gauge, and also as a function of the temporal offset between the time series, which could be as large as the TOPEX repeat period of 9.9 days. Three types of correlation were identified.

First, the time series with zero spatial separation had a correlation that varies slightly with the temporal lag (Fig. 4). This correlation is attributed to errors in the tide gauge data that are correlated in time. The fact that the maximum correlation, 0.20, is relatively small implies that the noise in the difference series is largely due to altimeter noise or to noise in the tide gauge data that is nearly uncorrelated in time. The best-fit functional form of this component of the correlation function is
ρmntmn
where tmn is the temporal offset between the difference series in days. Again, this component of the correlation function applies to series with zero spatial separation, which are the series derived from a single tide gauge but differing TOPEX passes.
The second type of correlation identified occurs when the temporal separation was small; specifically, when it was less than 0.2 days. This situation occurs when different tide gauge stations are crossed by the altimeter on the same, or adjacent, passes. This is interpreted as reflecting along-track persistence in altimeter errors. This interpretation is supported by the fact that difference series from closely spaced tide gauges that are associated with different altimeter passes showed no significant correlations even when the time differences were less than a day. This correlation was fit as a function of the distance between the stations (Fig. 5), although the time difference could be used as easily. The functional form of this component of the correlation is
ρ"mndmn2
where dmn is the distance (km) separating the series.
The third type of correlation discovered has a sinusoidal, once per revolution of the earth, dependence that decays over approximately four revolutions. The source of this correlation is not clearly understood, but the once-per-revolution dependence suggests a small remaining TOPEX orbit error. These correlations are quite small (less than 0.07) but are interesting in that these can introduce slight negative correlations. The best-fit functional form in this case is
ρmnωtmntmn2
where ω is 81.1 d−1, the frequency at which TOPEX orbits the earth, and tmn is again the time difference in days between the series. The final form used for the estimate of the theoretical correlation function is then
ρmnρmnρ"mnρmn
where the terms on the right-hand side of (9d) are defined in (9a–c).

After subtracting the estimate of the ρmn function given by (9d), the correlations (rmn) computed from the time series were reexamined. The distribution is now highly symmetric and is indistiquishable from a Gaussian distribution, which is expected if (9d) is a reasonable estimate of the theoretical correlation. It is likely that additional contributions to ρmn remain unaccounted for, but it is considered unlikely that the magnitude can be large enough to significantly affect the calculation of the weights using the formalism described above.

5. Results and discussion

Using the variances (σ2n) and correlation function (9d) obtained in the last section, the procedure outlined in section 3 was carried out to compute the time series of the drift estimate from (7) and its variance from (8). The result is shown as the solid circles and vertical error bars in Fig. 6. The error bars shown are ±1σ, as computed from (8), and are typically 5–6 mm. A large drift, particularly over the last half of the record, is readily apparent, along with a low-frequency oscillation in the first half of the time series.

Before discussing this result further, it is natural to ask how sensitive the result is to the specific correlation function used. To test this, all of the parameters in the various components of the fitted correlation function were increased and decreased by a factor of 2, and changes in the drift time series and the associated variances were monitored. The changes in the variances of the drift estimates were found to be less than 5% of the typical values. Also, the variances from (8) were only 10% larger than those obtained by assuming that the theoretical correlation function was equal to the identity matrix, that is, by assuming that all of the difference time series were independent. On the other hand, if we assume further that all the stations should receive equal weight, rather than taking into account the variance between TOPEX and the tide gauge at each site, then the variance of the drift estimates is almost 60% larger. We also note that using simple averages of all available data, rather than selecting the best series as described in section 2, results in variance estimates for the drift series that are nearly doubled. We conclude that choosing tide gauge stations where the differences between the TOPEX heights and the tide gauge sea levels are small is essential to getting the variance estimates correct, but that the consideration of the correlations in the difference time series is less important, although not completely insignificant.

Returning to the computed drift estimates (Fig. 6), recall from the earlier discussion that the TOPEX data were not corrected for the recently discovered algorithm error. The slow drift due to this error is shown in Fig. 6 with a solid curve. Note that no fitting has been done, although the means have been removed from both time series, which is necessary given that the tide gauge relative sea levels can track only temporal changes in the TOPEX height. It is obvious that the tide gauge analysis tracks the algorithm error rather well. Furthermore, the standard deviation of the difference between the tide gauge estimate of the TOPEX drift and the “truth” is 6 mm, as compared to 5–6 mm from the internal estimates of the uncertainty from the analysis. The ability of the tide gauge analysis to not only track the drift but also to produce an apparently reliable estimate of its ability to do so is extremely encouraging.

The recently discovered algorithm error has provided an invaluable way to evaluate the ability of the tide gauge analysis to monitor the stability of the TOPEX data and, because the technique is general, the stability of other altimeters as well. While this analysis is based on a repeat track analysis of the TOPEX data, this is not necessary and similar calculations using a crossover analysis have been done by other researchers (Murphy et al. 1996a). The algorithm error has also served to verify that accurate error estimates for the drift time series can be computed as well. For the final presentation of the TOPEX stability, the analysis was repeated using TOPEX heights that have been corrected for the algorithm error (Fig. 7). The primary thing to note is that the tide gauge analysis implies that the TOPEX data, after correction for the algorithm error, are stable to better than 10 mm over nearly 4 years. This level of stability is more than adequate for all but the most demanding applications of the TOPEX data. The most obvious example of such a demanding application is the computation of temporal changes in the globally averaged sea level height, a calculation that has been undertaken by several groups (e.g., Rapp et al. 1994; Nerem 1995; Minster et al. 1995).

Having noted the excellent stability of the corrected TOPEX data, it is also noteworthy that this analysis suggests that there is a significant remaining drift in the TOPEX data (Fig. 7). A trend fit to these drift estimates yields a value of −2.6 ± 0.6 mm yr−1. At least several explanations for the remaining drift exist. First, another as yet unrecognized source of drift might be present in the TOPEX heights. Second, it has been suggested (R. Cheney 1996, personal communication) that the assumption in the tide gauge analysis that the drift is globally uniform might not be valid. For example, if the drift is due to errors in an environmental correction (e.g., path delay due to water vapor), then the drift rate may be different in different parts of the world. If this is the case, then the geographical distribution of the tide gauges could be problematic. Third, the drift may be due to a systematic error in the tide gauge analysis. An obvious possibility is that the averaged tide gauge sea levels are contaminated by land motions that do not average to zero.

The possibility of a spatially varying drift rate was evaluated by repeating the tide gauge analysis using only gauges within 15° of the equator, which results in a drift rate of −3.7 ± 1.0 mm yr−1, and using only gauges more than 15° from the equator, which results in a drift rate of −1.2 ± 1.0 mm yr−1. These values are only different from the value obtained using all data (−2.6 ± 0.6 mm yr−1 at the 1σ level, but the larger value from the equatorial stations as opposed to the higher-latitude ones is consistent with an interpretation of a meridionally varying drift rate, possibly due to time-dependent errors in the water vapor correction. Statistically, the result is not significantly different from zero, and regardless of the subset of stations used, the negative trend remains, but the dominance of tropical stations in this analysis may lead to an overestimate of its magnitude. Lacking any quantitative way to estimate this bias, we will not correct our estimates of the remaining drift for this effect. But this check certainly suggests an obvious direction for future work. If a closer evaluation of the water vapor correction were to yield a meridional structure function for this potential error, for example, it might be possible to obtain improved estimates of the remaining drift.

The third possibility, that land motion at the tide gauges may be biasing the results, requires careful consideration. At the level of 1 mm yr−1, the point has certainly been reached where the drift inferred from the tide gauge analysis could be due to land motion at the gauge sites. This source of error has always been problematic for sea level rise calculations from the global tide gauge network (e.g., Douglas 1991). Two checks were done to evaluate this possibility. First, estimates of land motion due to postglacial rebound (Mitrovica et al. 1994) were examined. At the tide gauge sites used in this study these trends are approximately an order of magnitude too small to account for the observed trend in the TOPEX minus tide gauge time series. This primarily reflects the fact that this analysis is dominated by tide gauges from the Tropics and from islands that are not strongly affected by postglacial rebound. A second check was done by examining the long-term sea level trends at stations that have at least 5 years of sea level data and also have relatively small sea level variability (variance less than 100 cm2). For these series, which include 115 of the 149 difference series used in this analysis, the sea level trend can be computed with some precision. Taking these sea level trends and averaging them using weighting analogous to that used in deriving the drift estimate results in a “global” trend of 1.5 mm yr−1. Depending on how many of the stations can be considered independent, this average has a standard deviation of 0.5–1 mm yr−1. Note carefully, though, that this is computed from the sea level data only, and the rate is not expected to be zero. If the expected rate is taken to be the 1.8 mm yr−1 sea level rise rate determined by Douglas (1991), who used many more tide gauges and much longer time series, then the best estimate of the systematic error due to the choice of tide gauges is −0.3 mm yr−1. An uncertainty of 1 mm yr−1, the upper limit estimated for the averaged rate, will be assigned to the estimate.

Combining the estimate of the remaining drift (−2.6 mm yr−1) with the estimate of the systematic error (−0.3 mm yr−1) yeilds a drift estimate of −2.3 ± 1.2 mm yr−1, where the uncertainty is computed by combining the random error (0.6 mm yr−1) with the systematic error estimate (1 mm yr−1). Assuming that the errors have been fairly estimated, the remaining trend is too large to account for with the uncertainty in the tide gauge analysis. The uncertainty in the trend is dominated by the uncertainty involved in determining the potential systematic effect of land motion at the tide gauges. If the tide gauge heights were known in an absolute coordinate system, this error could be greatly reduced. This would require the use of GPS receivers to monitor the land motion at the gauges, which is becoming feasible (Ashkenazi et al. 1993; Carter 1994).

In conclusion, it should be reemphasized that the tide gauge analysis indicates that the present TOPEX sea surface height dataset is stable to at least 2 mm yr−1, which is a remarkable achievement. The recently discovered algorithm error confirms the ability of the tide gauge analysis to accurately monitor drift in the altimeter heights and additionally verifies that the internal error estimates for the drift time series are accurate as well. In some ways the latter is the more important finding. For the most demanding applications of the TOPEX data, such as the global sea level change calculations used as an example here, questions still remain, however, and further progress will require better knowledge of the vertical stability of the tide gauges. Finally, although this paper has focused on an application of this technique to the TOPEX altimeter, the method is in fact completely general and could be applied to any altimeter as long as the tide gauge network remains in place.

Acknowledgments

The original idea of using the tide gauge sea levels to produce a datum for altimetric sea surface heights data arose in conversations with Professor Klaus Wyrtki nearly 10 years ago, and the refinements to the technique since our original efforts benefitted greatly from many talks with him. The work described above has also been much improved by conversations with my colleagues on the TOPEX/Poseidon Science Working Team, and significant improvements were made following comments by two anonymous reviewers. Sea level data were obtained from the University of Hawaii Sea Level Center operated by Dr. Mark Merrifield, and assistance with data processing was provided by Shikiko Nakahara and David Stone at the University of Hawaii. This work was supported by NASA through the Jet Propulsion Laboratory as part of the TOPEX Altimeter Research in Ocean Circulation Mission.

REFERENCES

  • Ashkenazi, V., R. Bingley, G. Whitemore, and T. Baker, 1993: Monitoring changes in mean sea level to millimeters using GPS. Geophys. Res. Lett.,20, 1951–1954.

    • Crossref
    • Export Citation
  • Barlow, R., 1989: Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences. John Wiley and Sons, 204 pp.

  • Beers, Y., 1962: Introduction to the Theory of Error. Addison–Wesley, 66 pp.

  • Carter, W., Ed., 1994: Report of the Surrey workshop of the IAPSO tide gauge bench mark fixing committee. NOAA Tech. Rep. NOSOES0006, 81 pp. [Available from National Ocean Service, 1305 East-West Highway, Silver Spring, MD 20910.].

  • Cheney, R., B. Douglas, and L. Miller, 1989: Evaluation of Geosat altimeter data with application to tropical Pacific sea level variability. J. Geophys. Res.,94, 4737–4748.

    • Crossref
    • Export Citation
  • ——, L. Miller, R. Agreen, N. Doyle, and J. Lillibridge, 1994: TOPEX/Poseidon: The 2-cm solution. J. Geophys. Res.,99, 24555–24563.

    • Crossref
    • Export Citation
  • Christensen, E., and Coauthors, 1994: Calibration of TOPEX/Poseidon at Platform Harvest. J. Geophys. Res.,99, 24465–24486.

    • Crossref
    • Export Citation
  • Dillon, W., and M. Goldstein, 1984: Multivariate Analysis: Methods and Applications. John Wiley and Sons, 587 pp.

  • Douglas, B., 1991: Global sea level rise. J. Geophys. Res.,96, 6981–6992.

    • Crossref
    • Export Citation
  • Hayne, G., S. Hancock III, and C. Purdy, 1994: TOPEX altimeter range stability estimates from calibration mode data. TOPEX/Pos. Res. News,3, 18–20.

  • Minster, J.-F., C. Brossier, and P. Rogel, 1995: Variation of the mean sea level from TOPEX/Poseidon data. J. Geophys. Res.,100, 25153–25161.

    • Crossref
    • Export Citation
  • Mitchum, G., 1994: Comparison of TOPEX sea surface heights and tide gauge sea levels. J. Geophys. Res.,99, 24541–24553.

    • Crossref
    • Export Citation
  • Mitrovica, J., J. Davis, and I. Shapiro, 1994: A spectral formalism for computing three-dimensional deformations due to surface loads. Part II: Present-day glacial isostatic adjustment. J. Geophys. Res.,99, 7075–7101.

  • Morris, C., and S. Gill, 1994: Evaluation of the TOPEX/Poseidon altimeter system over the Great Lakes. J. Geophys. Res.,99, 24527–24540.

    • Crossref
    • Export Citation
  • Murphy, C., P. Moore, and S. Ehlers, 1996a: Preliminary study of the use of tide gauge data to unify altimetry data from separate satellites. Adv. Space Res., in press.

  • ——, ——, and P. Woodworth, 1996b: Short-arc calibration of the TOPEX/Poseidon and ERS-1 altimeters utilizing in situ data. J. Geophys. Res.,101, 14191–14200.

    • Crossref
    • Export Citation
  • Nerem, S., 1995: Global mean sea level variations from TOPEX/Poseidon altimeter data. Science,268, 708–710.

    • Crossref
    • Export Citation
  • Rapp, R., Y. Yi, and Y. Wang, 1994: Mean sea surface and geoid gradient comparisons with TOPEX altimeter data. J. Geophys. Res.,99, 24657–24667.

    • Crossref
    • Export Citation
  • Sciremammano, F., Jr., 1979: A suggestion for the presentation of correlations and their significance levels. J. Phys. Oceanogr.,9, 1273–1276.

    • Crossref
    • Export Citation
  • Wyrtki, K., and T. Bongers, 1987: Sea level at Tahiti—A minimum of variability. J. Phys. Oceanogr.,17, 164–168.

    • Crossref
    • Export Citation
  • ——, and G. Mitchum, 1990: Interannual differences of Geosat altimeter heights and sea level: The importance of a datum. J. Geophys. Res.,95, 2969–2975.

    • Crossref
    • Export Citation
Fig. 1.
Fig. 1.

Illustration of how the TOPEX and tide gauge time series were paired. The open circle is the location of one of the tide gauges (at Papeete in French Polynesia) used in this study. The small dots mark the locations where TOPEX sea surface height time series are available. The larger solid dots mark the four TOPEX time series used to form difference series with the tide gauge sea level record. The TOPEX points lie at the same latitude rather than at the point of closest approach, but that is not critical to the results. Points at the same latitude are chosen because the meridional length scales of sea level variability are significantly smaller than the zonal scales for tropical stations, which form a large fraction of the available tide gauge locations.

Citation: Journal of Atmospheric and Oceanic Technology 15, 3; 10.1175/1520-0426(1998)015<0721:MTSOSA>2.0.CO;2

Fig. 2.
Fig. 2.

Locations of tide gauges used in this study. The locations of the 53 tide gauges remaining after the quality control checks described in the text are completed and are marked by solid circles. Note that each tide gauge can produce up to four difference time series (see Fig. 1), so there are fewer stations shown than the number of difference series (149) used in the final set of analyses.

Citation: Journal of Atmospheric and Oceanic Technology 15, 3; 10.1175/1520-0426(1998)015<0721:MTSOSA>2.0.CO;2

Fig. 3.
Fig. 3.

Histogram of the sample correlations (rmn). Estimates were computed for every possible pairing of the 149 difference time series. Typically there are order 60 degrees of freedom, and the standard deviation of the correlation estimate assuming zero true correlation is approximately 0.13 (see text).

Citation: Journal of Atmospheric and Oceanic Technology 15, 3; 10.1175/1520-0426(1998)015<0721:MTSOSA>2.0.CO;2

Fig. 4.
Fig. 4.

The fitted correlation function for the case of zero spatial separation. This component of the correlation function applies to difference time series derived from the same tide gauge but different TOPEX passes. The solid curve is the fitted function given by (9a) in the text. The solid dots and errors bars are bin-averaged correlation estimates derived from the data. The bins are chosen to have equal numbers of points in each. Note that the fitting was done with the actual data values, and the bin averaging is done only to aid the presentation of the result.

Citation: Journal of Atmospheric and Oceanic Technology 15, 3; 10.1175/1520-0426(1998)015<0721:MTSOSA>2.0.CO;2

Fig. 5.
Fig. 5.

As in Fig. 4 but for the case of small temporal separations that arise when the altimeter passes over two different tide gauges on the same orbit.

Citation: Journal of Atmospheric and Oceanic Technology 15, 3; 10.1175/1520-0426(1998)015<0721:MTSOSA>2.0.CO;2

Fig. 6.
Fig. 6.

Result of the initial analysis. The solid circles and error bars are the result of the TOPEX drift analysis using the tide gauge data. At this point the TOPEX data have not been corrected for the recently discovered algorithm error (shown as the solid line), which was in fact not known at the time the tide gauge analysis was done. The means have been removed from both time series because the tide gauge mean is arbitrary, but otherwise, no fitting has been done.

Citation: Journal of Atmospheric and Oceanic Technology 15, 3; 10.1175/1520-0426(1998)015<0721:MTSOSA>2.0.CO;2

Fig. 7.
Fig. 7.

Final result of the tide gauge determination of the TOPEX stability. The solid dots and error bars show the results of the tide gauge estimates of the TOPEX drift, as in Fig. 6, but the TOPEX data have now been corrected for the algorithm error, and the tide gauge analysis has been recomputed.

Citation: Journal of Atmospheric and Oceanic Technology 15, 3; 10.1175/1520-0426(1998)015<0721:MTSOSA>2.0.CO;2

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