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The final set of corrections to be discussed is a set of adjustments that must be made to properly interpret the satellite height measurement, which is computed from an average of many time delay estimates from within the satellite "footprint." This footprint is of order several tens of kilometers squared, and it is not difficult to imagine that the satellite average obtained once every 10 days or so can differ from the average that we would obtain if we had continuous in situ measurements of sea level over the entire area of the footprint at every instant in time rather than a single sample each time the satellite flies over. If there is such a difference we term this a surface effect, and there are three such effects that we will discuss. These effects are referred to as the sea-state bias, the tides, and the inverted barometer adjustment.

The first of these surface corrections, the sea-state bias, arises because the surface of the ocean is rarely perfectly smooth, but is instead nearly always covered by capillary waves and surface gravity waves. The satellite footprint is large enough that in principle these waves will average out to nearly zero over that spatial scale, but this does not work in practice. One reason for this is that the troughs of the waves reflect the radar energy more effectively than the wave crests, leading to a bias in the return to the satellite. That is, we get more reflection from wave troughs than crests and when the average is made it is found to be lower than the mean height that would be obtained if no waves were present. There are actually several distinct effects involving biases due to surface waves, but for simplicity we will combine these all under the term sea-state bias.

To do the best correction for this bias we would need to know the wave heights in the footprint as a function of the wavelength of the waves; that is, we need to know the complete wave spectrum. In fact, we can measure the wave height (SWH) from the T/P radar pulses, but the spectrum is not available to us. To obtain a proxy estimate of the effect of the wave spectrum, wind speed is also needed (Fu and Glazman, 1991) in an analysis of the sea-state bias. The idea is that when the wind is blowing, the waves tend to be short and choppy, while during calm conditions the dominant waves are relatively long swells from further away. The correction for the sea-state bias is thus parameterized in terms of the SWH and wind speed and the parameters of the model are obtained by fitting to the T/P data itself. The details of this procedure are beyond our scope here, but the interested reader is referred to Gaspar et al (1994).

The ocean tides give rise to another surface effect that must be accounted for, but of a very different sort. The problem here is that the altimeter observes a given point on its ground track very infrequently, once every 9.9 days for T/P, in comparison to the dominant tidal frequencies, which are once or twice per day. Therefore, each time the altimeter passes over this point, the tidal signal is at a completely different phase, meaning that it may be high tide one time, low tide at another time, or anywhere in between. So we have an aliasing problem in which a signal of one frequency is not sampled often enough and ends up masquerading as a lower frequency signal. For example, the M2 tidal component at a period of 12.42 hours appears in the T/P data at a period of about 62 days. For readers unfamiliar with the aliasing problem, a good discussion can be found in Bloomfield (1976). Aliasing is a significant problem for altimetry because the ocean tides have surface deviations of order 1 m, compared to a few centimeters for the low-frequency ocean signals that we wish to observe. If these aliased tidal signals could not be removed, it would be difficult, if not impossible, to make proper interpretations of the sea surface height time series obtained from the altimeter.

To deal with the tides it is necessary to predict the tides using a model and to remove this estimate from each of the altimetric height measurements. In practice, since the tidal alias periods are easily computed from the motions of the sun and moon and the rotation of the earth, the T/P data itself can be used to make empirical tidal models once long enough altimetric time series have been obtained. In addition, hydrodynamical tidal models can be used, especially if in situ and T/P data can be assimilated. These tidal modeling efforts have been extremely successful during the T/P mission, and the tide models derived from T/P are now the best available in large parts of the world's oceans. For more complete information about estimating and correcting for tides in the T/P data, see Molines et al (1994).

The final surface correction to be discussed is called the inverted barometer correction. This should probably be referred to as an adjustment rather than a correction, however, since we are not correcting any error in this case. To understand what this adjustment is, consider that the altimetric observation is simply a geometric measurement of the height of the sea surface from the reference ellipsoid. What oceanographers often need, however, is an estimate of the pressure deviation due to the sea surface height's deviation from its long-term mean value. Such pressure estimates can be used to infer surface circulation, for example, from the geostrophic relationship. In estimating the pressure due to the water deviation, however, we need to adjust for any pressure fluctuation due to increased or decreased atmospheric load as measured by surface atmospheric pressure. If it is desired to make this conversion from sea surface height to ocean surface pressure deviations, one must add in the atmospheric pressure deviation, which is referred to as making the inverted barometer correction, or adjustment. In essence we obtain the pressure field that the subsurface ocean would experience if the atmospheric pressure field were constant. The atmospheric pressure fields used for this adjustment are the same as used for the dry tropospheric correction to the range discussed earlier.

Making an inverted barometer adjustment is not so relevant to studies aimed at determining sea level change, since sea level is also a geometric measurement and is therefore analogous to the altimetric sea surface height measurement without the inverted barometer adjustment applied. We do not make this adjustment in most of the work that we will present here, but it is used almost routinely in studies of ocean circulation. We do, however, apply this correction in some of our calculations of regional sea level change in order to reduce "noise" due to regional scale atmospheric variations.

In the case of the surface corrections, particular attention must be given to these errors since all of the corrections rely on model output rather than direct observations. The inverted barometer adjustment is particularly noisy because of numerical weather model errors in the surface atmospheric pressure fields, which is another reason we do not generally apply this correction in our work. The tide models are not perfect, but there seems little chance of these errors introducing low-frequency drift errors. The tidal signals and the tide model estimates both occur at discrete frequencies, implying that the tide errors do also. Since T/P was designed to keep large tidal components aliased to relatively short periods (less than 1 year), these errors average out when estimating very-low-frequency trends. So tide model errors are not considered a significant source of drift error for T/P, although this may not be true for other altimetric satellites where the orbital parameters, and the associated tidal alias periods, are not chosen as carefully. The sea-state bias models, on the other hand, are difficult to assess in terms of the potential for low-frequency drift and such a possibility has to be kept in mind.

In each of the preceding sections we have described the basic components of the altimetric measurement system, and have given a very brief assessment of the possibility that each of the components of the system might give rise to low-frequency errors, or drifts. In this section we will assemble more information about the errors in the overall system, again giving special attention to the likelihood that any of these errors could bias estimates of sea level trends computed from the altimetric heights. As discussed at the beginning of the chapter, these are the types of errors that would be most damaging to any attempt to estimate sea level change rates from altimetry. For each of the components of the system we will cite the initial geophysical evaluations of the T/P data, which have been summarized by Fu et al. (1994). For the orbit determination and the tides we will note more recent evaluations because significant improvements have occurred in these areas.

Starting with the range measurement, evaluations done with the first 2 years of T/P data indicated that the range errors were of order 2 cm, not including uncertainties in the correction for the atmospheric index of refraction. The long-term stability of the range measurement, as fixed by the internal calibration, is still an area of debate. The overall size of the internal calibration correction, however, indicates that the stability is good to at least several millimeters per year, a remarkable achievement even if it may not be quite up to the standard needed for estimating global sea level change.

For the orbit determination, the prelaunch requirement was that the orbit should be determined to about 10 cm. In fact, the T/P science teams charged with computing precise orbits had already exceeded this standard by the time of the launch; the initial estimates of the orbit errors were approximately 3.5 cm, with less than 2 cm of this in the time-independent error mentioned above. The computed orbits have continued to improve since then and present error estimates are on the order of 2-3 cm for a typical global value (Marshall et al., 1995). This ongoing improvement of the quality of the orbital calculations is one of the true achievements of the T/P mission, and many of the scientific studies being undertaken with these data would not be possible had this not happened. As discussed above, it is not expected that the orbit errors will lead to significant drift errors, and there is no evidence of such errors to date. As mentioned earlier, reference frame stability could become a matter of concern over several decades and multiple missions, especially since the satellite tracking techniques (on which the reference frame is based) are continually evolving.

The situation with the atmospheric corrections, which essentially result in estimates of the effective index of refraction for the atmosphere, is a bit more complicated, particularly with regard to the wet tropospheric correction. The initial T/P evaluations led to estimates of 0.5 cm for the error in the ionospheric correction, 0.7 cm for the dry tropospheric correction, and 1.1 cm for the wet tropospheric contribution to the range correction. The drift error in the wet correction that we discussed earlier is, however, worrisome. Although to our knowledge the drift error is now corrected (Keihm et al., 1998), it is a correction that obviously has the potential to produce significant trend errors and must therefore be watched closely in the future and in other altimetric missions.

Finally, for the surface corrections, the initial error estimates from the summary by Fu et al. (1994) were about 2 cm for the sea-state bias and 5 cm for typical global tide model error. As with the orbits, the quality of existing tide models was much better than anticipated before launch, and the models continue to improve. It is now estimated that the tide model errors are approximately 1-2 cm (Shum et al., 1997) in the open ocean, and as we have already stated, it is unlikely that these errors will introduce significant bias into long-term sea level trend estimates. The same appears to be true for the sea-state bias estimation, although this cannot really be established yet, particularly if the estimates of wind speed and significant wave height were to exhibit unrealistic low-frequency trends.

One way to make an overall assessment of the precision and accuracy of the T/P system for producing sea surface heights is to compare these heights to sea level measurements from tide gauges. Although we cannot easily attribute any errors so observed to a particular component of the altimetric system, such comparisons do provide an important end-to-end assessment of the total T/P system. A comparison of this type was carried out by one of us (Mitchum etal., 1994) shortly after the launch of the T/P satellite. Those results indicated that the overall precision of the T/P data was better than 4 cm at any given location using the highest frequency data available from the T/P system. Cheney et al. (1994) averaged the data to monthly time scales and obtained error estimates of order 2 cm, indicating that much of the error in the T/P system is uncorrelated temporally and therefore averages down like random noise. Even more encouraging is that the fact that in the period following these studies, the T/P data have improved, and similar calculations now show errors of less than 3 cm for the highest frequency data, which is almost a factor of 2 reduction in the error variance. An example of a T/P and tide gauge comparison is shown in Fig. 6.2, which shows the tide gauge record at Pohnpei in the central tropical Pacific and the analogous T/P-derived sea surface height series in the vicinity of the island. This comparison is better than the globally averaged result, but is not particularly unusual for island stations in the tropical parts of the oceans. As we will discuss in the next section, the quality of comparisons such as these allows us to make sensitive analyses of the errors in the T/P data and identify drifts smaller than 1 mm/yr.

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