## Calibrating The Measurement Calibration Overview

Since we are particularly concerned with identifying low-frequency errors, or drifts, in the T/P data when using these data for estimating sea level change, it would be most useful if we had a method for specifically estimating drifts from independent data. In fact, techniques to do this have been developed using the tide gauge measurements referred to above, as well as by using other satellite altimeters and other types of in situ data. Although our focus in this chapter will be on the use of the global tide gauge network, we will briefly review these various other methods as well. Of course, before any of these methods can be trusted one must have some confidence that the data used as a benchmark, for example the tide gauges, are themselves stable over long time scales. We will address this issue as well for the tide gauges, and the reader is also referred to the discussion of tide gauge data in Chapter 3.

Pohnpei (7°N, 158°E) sea level (solid) vs. TOPEX heights (dashed)

Pohnpei (7°N, 158°E) sea level (solid) vs. TOPEX heights (dashed)

Mitchum (1998) discusses various approaches to the problem of estimating the T/P drift error and categorizes the approaches that use in situ data according to the strategy adopted for minimizing the error of the drift estimate. This discussion can be summarized as follows. First, all of the methods proceed by computing a difference of the T/P heights with an in situ estimate of sea level. These in situ estimates can come from a single calibration site such as the offshore California Harvest oil production platform (e.g., Christensen et al., 1994), from tide gauges in lakes (Morris and Gill, 1994), from dynamic height computed from open ocean temperature profiles, or from the global tide gauge network (Mitchum, 1994; Mitchum, 1998). Computing differences is done in order that the ocean signals that are common to both the T/P and in situ data will cancel, isolating the errors for further analysis.

Regardless of the source and extent of the in situ data, in each case there are one or many locations where differences are computed but only some fraction N of these can be considered statistically independent. For simplicity in this illustration, we assume that at each location the standard deviation of the (T/P minus in situ differences) can be characterized by a single value that we will denote as a. Under these assumptions the variance of the global mean difference at any point in time, which results in a time series for the drift estimate, will be a2/N. The various approaches to the drift estimation problem can be classified according to the strategy adopted for minimizing this variance.

In the case of the single site calibration, such as is done at the Harvest platform, the strategy is to obtain the smallest difference between the T/P data and the in situ data, which are from a tide gauge, and thus to minimize a1. In the case of the global tide gauge analysis, larger a2 values are accepted, but TV is made as large as possible. A similar strategy characterizes the approach using dynamic height from temperature profiles, although the a2 values will be larger due to the more indirect method of estimating sea level. The use of tide gauges only in the Great Lakes is an intermediate approach, wherein a2 is larger than at Harvest but smaller than at many open ocean tide gauges, but N cannot be made as large as with the open ocean approach. To date, the global tide gauge approach has arguably yielded the most sensitive results for the time-dependent error because the increase in N has outweighed the slightly larger values of a2 at the various gauges used in the analysis. The global tide gauge approach successfully identified an algorithm error in the early T/P data and also gave the first indication of a possible drift in the wet correction (Mitchum, 1998). The Harvest platform analysis, on the other hand, being the only technique that can presently detect mean, or time-independent, biases is extremely valuable for determining offsets between different satellite missions. It is important to note, however, that efforts to improve all of these methods are active at present, and results should improve with time. In summarizing the global tide gauge approach to estimating T/P drift we will give a short review of the development of the method, and also present the most recent results from this approach. But first we will briefly consider the stability of the tide gauges themselves, since assuming that these instruments are stable is a key assumption of the method.

### 6.3.2 Using Tide Gauges for Calibration

One of the advantages of using tide gauge data for calibration is that methods have long been in place to maintain the stability of the vertical reference point for tide gauges. Some discussion of these methods is given in Chapter 3, and more details are available elsewhere (e.g. Pugh, 1987). One of the key components of the tide gauge system is the tide staff, which is simply a calibrated pole placed close to the tide gauge and read periodically by a human observer. Comparing gauge and staff readings allows the tide gauge operator to identify drifts in the mechanical tide gauge system, since the very simple and direct staff measurements, although noisy, should not be subject to serious drift problems. But these staff measurements have often been questioned, and an illustration of the quality of the staff readings will be given here to provide some confidence in these measurements.

Our example is drawn from the Yap tide gauge station in the western tropical Pacific. This station was chosen because we have over 15 years of contemporaneous gauge and staff readings where the staff readings have never been used to correct the tide gauge data, thus keeping the two data sets completely independent. The staff readings are taken several times a week and do not resolve the ocean tides, so the staff heights were fit to a simple tide model using a least-squares criterion. The residuals from this tide fit were then averaged to monthly values. These monthly values are compared to the monthly mean gauge readings in the upper panel of Fig. 6.3, which show that there is very good agreement between these two time series at low frequencies. This agreement can be quantified as a function of frequency by computing a cross spectral analysis between these two time series, the results of which are shown in the bottom two panels of the same figure. The response function measures the ratio of the amplitude of the signals observed by the staff to those observed by the gauge. At periods longer than about a year, this function is very near unity, meaning that the staff successfully recovers the amplitude

Gauge (solid) and staff (dashed) sea levels at Yap Island

Gauge (solid) and staff (dashed) sea levels at Yap Island

Figure 6.3 Comparison of sea level from the tide gauge (solid line) and the tide staff (dashed line) at Yap Island. The upper panel shows the monthly mean time series, and the lower two panels are from a cross spectral analysis of these two time series. At the lower left is the response function (staff variance to gauge variance); at the lower right is the coherence-squared function. The dashed line in the coherence-squared panel gives the 95% probability point for the null test that the true coherence-squared is different from zero.

Figure 6.3 Comparison of sea level from the tide gauge (solid line) and the tide staff (dashed line) at Yap Island. The upper panel shows the monthly mean time series, and the lower two panels are from a cross spectral analysis of these two time series. At the lower left is the response function (staff variance to gauge variance); at the lower right is the coherence-squared function. The dashed line in the coherence-squared panel gives the 95% probability point for the null test that the true coherence-squared is different from zero.

of the sea level variability at those periods. In the lower right panel, the coherence squared function, which is basically a correlation squared as a function of frequency, is plotted, and at the same subannual frequencies the coherence values approach a value of 1, indicating nearly perfect agreement. The dashed line on this panel shows the critical coherence value such that values larger than this can be considered significantly different from 0 at the 95% confidence level. From the analysis it is clear that the staff measurements, despite being noisy and infrequent are faithfully recording the sea level variations, and with a system that is very unlikely to be drifting. Although this analysis at one station is not definitive, it gives some confidence that the use of tide staffs in maintaining the low-frequency stability of tide gauges is a viable and reliable approach.

There is, however, a much more serious problem with the tide gauge data that the staff data cannot address. Both the tide gauge and the staff measure sea level as a difference of the ocean height and the land on which the instruments sit. The stability of these instruments is monitored relative to the land near to the gauge by periodic surveying and in principle one can determine if there is instability in the land very close to the gauge. If the land is moving vertically on a larger scale, however, none of these measurements will detect it. This means that the sea level trend computed from tide gauge data must be considered to be the difference between the ocean sea surface trend and the vertical land motion rate. And thus a trend in the difference between the T/P data and the sea level from a given tide gauge must be considered as the sum of the T/P drift rate and the land motion rate.

A striking demonstration of the effect of land motion has been given by Cazenave et al. (1999) at the island of Socorro off the west coast of Mexico using independent estimates of the land motion at the site from DORIS measurements. For the purpose of demonstrating the problem here we have plotted the long time series of sea level obtained at two locations along the Hawaiian Ridge in the central subtropical Pacific. These locations are at Hilo on the south end of the main Hawaiian Island chain, and Honolulu at approximately the middle of the main Hawaiian Islands (Fig. 6.4). It is clear at a glance that the sea level rise rates at the two stations are significantly different even though the stations are quite close together. The interpretation of these results is that the Hilo tide gauge, which is located in an area of active volcanism, is subject to larger land motion than the gauge at Honolulu. Other examples of this sort are easily found, as discussed by Douglas in Chapter 3. The only ultimate solution to the land motion problem is to have space geodetic measurements, such as DORIS or GPS, at any tide gauge used in the drift estimation. But at present very few gauges are so equipped, and alternative methods for estimating land motion and assessing the uncertainty due to land motion have been devised. These methods will be briefly summarized at the end of this section, but first we will review the development of the basic methods used to estimate T/P drift from the global tide gauge network.

Honolulu and Hilo records - note difference in rise rate

Honolulu and Hilo records - note difference in rise rate

### 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Figure 6.4 Sea level trends at Honolulu and Hilo in the Hawaiian Islands (after Nerem and Mitchum, 1999). The time series are annual mean sea levels and the rates quoted are from standard least-squares fits. The Hilo station is in an area of active volcanism.

### 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Figure 6.4 Sea level trends at Honolulu and Hilo in the Hawaiian Islands (after Nerem and Mitchum, 1999). The time series are annual mean sea levels and the rates quoted are from standard least-squares fits. The Hilo station is in an area of active volcanism.

The initial suggestion of using the tide gauge data as an assumed stable check for altimetric heights was given by Wyrtki and Mitchum (1990) in an analysis of Geosat altimeter data. This was followed by a more sophisticated, but still preliminary application of this approach to the first 2 years of T/P data (Mitchum, 1994). As the length of the T/P time series increased, the method was improved significantly. The discovery in the T/P data of an algorithm error that was successfully captured by the tide gauge analysis (Mitchum, 1998) resulted in a much wider acceptance of these estimates of the T/P altimeter stability. The method outlined by Mitchum (1998) matched T/P data to tide gauge data by simply choosing the T/P data at the nearest point of approach at the latitude of the tide gauge. Also, only the nearest four ground tracks were examined for each tide gauge. But a careful statistical analysis was done to assess which T/P and tide gauge difference series could be considered independent, and to determine an optimal way in which to combine the difference series from different tide gauges together in order to arrive at a globally averaged difference curve that can be interpreted as the T/P measurement drift rate. The problem with land motion was recognized, but at that time only a crude estimate of the magnitude of the potential bias error due to land motion was made. It was estimated that the net effect of land motion after averaging over the global network was of order 0.4 mm/yr, but with a larger uncertainty of order 1 mm/yr. This term in fact dominated the error estimate for the T/P drift rate. The method derived in the Mitchum (1998) paper results in the drift series shown in the top panel of Fig. 6.5. Subsequent improvements made to the method will be addressed now, and are the topic of a paper now being prepared for publication.

There were three obvious ways in which the method of Mitchum (1998) could be significantly improved. First, that paper raised the issue of a possible drift in the wet troposphere correction. Such a drift was subsequently identified by Keihm et al. (1998) and a modified correction proposed. This modified wet correction was incorporated into all of the subsequent estimations of the T/P drift. Second, the method that Mitchum (1998) used to match the T/P data to the tide gauge data was not necessarily optimal for eliminating ocean signals because there was no allowance for spatial or temporal lags. Third, no attempt to correct for land motions was made at all.

Original method using cycles 1-212

Original method using cycles 1-212

Figure 6.5 T/P drift estimates from the original method (upper panel) given by Mitchum (1998) and from the improved method outlined in Section 3. Each point is an average over the global tide gauge network during a 9.9-day T/P cycle, and the vertical bars are the error estimates for each cycle estimate of the drift series.

Figure 6.5 T/P drift estimates from the original method (upper panel) given by Mitchum (1998) and from the improved method outlined in Section 3. Each point is an average over the global tide gauge network during a 9.9-day T/P cycle, and the vertical bars are the error estimates for each cycle estimate of the drift series.

The second possible improvement—making a better ocean signal cancellation by matching up the T/P and tide gauge data more precisely—is necessary because the T/P and tide gauge data are not taken at precisely the same location. Since the T/P ground track spacing can be nearly 300 km, there can be significant temporal lags between the two time series in areas where the ocean signals are dominated by slowly propagating ocean waves. An example of this was given by Mitchum (1994) at the tide gauge station at Rarotonga. In such cases lagging the series appropriately results in a better cancellation of these ocean signals, and consequently a smaller variance in the differences, which in turn allows a more sensitive determination of the T/P drift rate. Similarly, simply taking the T/P data at the same latitude as the tide gauge, as was done by Mitchum (1998), does not allow for spatial deformations in the ocean signals. The solution to both of these problems is to define the "best" T/P data to use with the tide gauge sea levels for given temporal and along-track spatial lags, with these lags chosen so that the variance of the difference time series is minimum.

Making this modification to the original method and using the modified wet correction result in the time series shown in the bottom panel of Fig. 6.5. Comparing this series to that in the upper panel, which uses the original method, shows that the magnitude of the linear trend is somewhat reduced. This is due to the inclusion of the modified wet correction. In addition, the point-to-point variability in the time series is much smaller on the lower panel as compared to the upper. This reduction is from the improved cancellation of the ocean signals due to the better matching of the T/P and tide gauge time series. The reduced noise level in the drift series opens up the possibility that smaller trends might be detected and also that shorter term errors might be more reliably detected, providing a generally more useful tool for studying the T/P drift characteristics.

Figure 6.5 shows that improving the wet correction and allowing for spatial and temporal lags in the ocean signals observed by the two measurement systems significantly improve the drift estimate. There is, however, still the larger problem of what to do about land motion at the gauges and how to estimate the uncertainty in the T/P drift estimates due to these motions. As stated earlier, the ultimate solution to this problem awaits the availability of long time series of space geodetic information at each gauge, but in the interim alternatives have been devised and are continuing to be improved. In brief, at each gauge two estimates of land motion are made, one from the tide gauge data and one from the nearest geodetic measurements by either GPS or DORIS. Uncertainties for each of these estimates are also evaluated and the two are then combined to make the best estimate possible for each tide gauge. The estimate based on the geodetic information is simply the land motion rate at the nearest continuous GPS or DORIS station within 50 km of a coastline. Of course, for the majority of stations the closest station is many hundreds of kilometers distant. If no geodetic rate was available within 500 km, then no geodetic estimate was attempted. For the other stations, the land motion rate estimate was set to the observed rate, but the uncertainty was inflated with a function that increased with distance from the tide gauge. For stations more than approximately 200 to 300 km away, the uncertainties become large enough that the geodetic rates essentially have no weight in the analysis.

The internal estimate of the land motion rate—the one derived from the tide data—is somewhat more complicated. In this case it is assumed that the sea level change rate computed from long time series of sea level, which in this case means at least 15 years and usually more like 25 to 40 years, is approximately the true ocean rise rate minus the land motion rate. Since we are forced to make these estimates from relatively short time series (see Chapter 3 by Douglas for warnings about attempting this calculation), we also tried to estimate interannual ocean signals associated with the EI Nino/ Southern Oscillation events when computing the rate estimates. Under this assumption, a land motion rate estimate and its uncertainty can be made from the apparent rate and its uncertainty and from an estimate of the true background rate, which for our present purposes is taken to be 1.8 mm/yr (Douglas, 1991; Chapter 3). Clearly this calculation is somewhat speculative, but since our hope is to remove the bias from the drift estimate only when averaging many stations together rather than to accurately estimate rates at individual stations, we expect that the results will still be an improvement over no land motion correction at all.

Although this discussion of the land motion correction is brief, it does give the basic idea. When these land motion corrections are applied, we obtain the drift time series shown in Fig. 6.6 in the upper panel. To get an idea of the net effect of applying these land motion estimates at the individual gauges, we differenced this time series from the previous calculation that included the improved wet correction and the better signal matching but did not make any adjustment for land motion (Fig. 6.6, bottom). The net result, not surprisingly, is mainly a change of the estimated trend. The sign of this change corresponds to net subsistence of the land at about 0.5 mm/yr when all of the global gauges are averaged. Because we have had to make a number of assumptions in creating this land motion correction, it is necessary to ask how sensitive the results are to these assumptions. The critical assumption turns out to be the rate taken for the true ocean rise rate. If this parameter is varied, the inferred T/P drift rate changes significantly (Fig. 6.7), and for a true rate in the range of 1-2 mm/yr, the range of the T/P drift rate estimates is approximately 0.8 mm/yr. From this we conclude that the uncertainty due to the land motion is of order 0.4 mm/yr. This is still the largest uncertainty in the drift estimate, but is a significant improvement over the previous estimate (Mitchum, 1998) of order 1 mm/yr.

Current method with land motion estimate included

Current method with land motion estimate included

1993 1934 1995 1996 1997 1998 1999 2000

Figure 6.6 The upper panel is as in Fig. 6.5, except that a land motion estimate has now been made for each tide gauge time series. See the text for the details on how this land motion correction is estimated. The lower panel is the difference between the upper panel of this figure and the lower panel on Fig. 6.5, which shows the effect of adding the land motion estimates. The error bars in the lower panel are computed under the assumption that the errors in these two times series are not independent.

1993 1934 1995 1996 1997 1998 1999 2000

Figure 6.6 The upper panel is as in Fig. 6.5, except that a land motion estimate has now been made for each tide gauge time series. See the text for the details on how this land motion correction is estimated. The lower panel is the difference between the upper panel of this figure and the lower panel on Fig. 6.5, which shows the effect of adding the land motion estimates. The error bars in the lower panel are computed under the assumption that the errors in these two times series are not independent.

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