Results of waveform retracking around Taiwan

We have experimented with waveform retracking around Taiwan using the retrackers described above. One important issue is to find the optimal retracker for the subsequent gravity anomaly derivation. The selection of such a retracker is based on the following two criteria:

a）success rate of retracking, and

b）the standard deviation of the differences between the retracked SSHs and modeled geoid heights.

The geoid model for use in Criterion 2 is a gravimetric geoid model of Taiwan computed from the latest land/marine gravity data and airborne gravity data (Hwang et al., 2005).

Because the assumption made in the OCOG retracker, it is only used for computing a priori values of the parameters needed in the other retrackers. Table 1 shows a typical comparison of retracked SSHs and geoidal heights along Geosat/GM track 85206 (data on day 206, 1985; see Fig. 5). In Table 1, we list the number of successfully retracked waveforms, the number of raw waveforms, the ratio between these two numbers and the standard deviation of the differences between retracked SSHs and geoidal heights for each retracker. Because of the complex waveforms near the coasts, the success rate of the Beta-5 retracker is only 70%. There is no problem of convergence for the threshold retracker, so its success rate is always 100%. The improved threshold retracker yields almost 100% of success rate. The standard deviations of the height differences in Table 1 are based on only retracked SSHs. Fig. 5 shows retracked SSHs and geoid heights along track 85206. The deviation of SSH from geoidal height increases as track 85206 approaches the land. In other tests that we have done, the Beta-5 retracker delivers success rates of retracking at about 70-80 %, which is considered inadequate for retracking around Taiwan. In terms of both success rate and standard deviation, Table 1 shows that the improved threshold retracker is significantly better than the other two retrackers. The results from other tests confirm that the improved threshold retracker indeed outperforms the other two retrackers, and hence it is selected as the optimal retracker in this paper.

The improved threshold retracker is further tested considering the dependency of retracking accuracy on region and depth. Fig. 6 shows the ground tracks of retracked

SSHs around Taiwan. In total, there are Geosat/GM 165 tracks here. The performance of the improved threshold retracker is assessed at four marine zones in Fig. 6 and at regions of different distances to the shores. Fig. 7 shows the distribution of the differences between retracked SSHs and geoidal heights. As seen in Fig. 7, retracking not only improves the accuracy of SSH over the coastal waters, but also over the open ocean. The places with most improved SSHs are near islands in the ocean and around barrier islands off the west coast of Taiwan. It is clear that not all SSHs can be properly corrected by retracking, as there are still remaining large errors seen in Fig. 7. This is a limitation of waveform retracking. These large errors may be caused by improper retracking and/or tide model error. Regarding tide model error, Hwang (1997) has assessed the accuracy of the CSR3.0 tide model around Taiwan and found that, the model error is the least (about few cm) in the Pacific Ocean east of Taiwan and the largest (about few decimeter) in the Taiwan Strait. At the central Taiwan Strait off the coasts of the mainland China and Taiwan, the tidal amplitude can reach 3 m. Here the difference in the M2 tide amplitude between CSR4.0 and observation is found to be about 50 cm. Efforts beyond waveform retracking for improving altimetry quality are presented in Anzenhofer et al. (1999).

The improvement percentage (IMP) of retracked SSHs is also computed, using

%

where σ_Raw and σ_Retracked are the standard deviations of the differences between raw SSHs and geoidal heights, and retracked SSHs and geoidal heights, respectively. Table 2 shows the improvement percentages in four cases. A negative improvement percentage indicates that retracking deteriorates SSHs. Table 2 shows that the improvement percentage increases as the waters approaches the land. While the difference in the improvement percentage between the marine zones at 20 km and 10 km to the shores is only marginal, the difference in standard deviation is significant. The largest improvement is at Zone 1 and the least improvement is at Zone 4, irrespective of the distance to the shore. However, SSHs in Zone 2 are degraded by waveform retracking, with unknown reason.

4. Gravity anomalies from retracked Geosat/GM altimetry 4.1. Gravity anomaly derivation by least-squares collocation

We have experimented with two methods of gravity derivation from retracked SSHs of Geosat/GM. One is the inverse Vening Meinesz formula (Hwang, 1998) and the other is least-squares collocation (LSC) (Moritz, 1980; Hwang and Parsons, 1995). In all

experiments (Section 4.2), LSC always outperforms the inverse Vening Meinesz formula by few mgal in the accuracy of computed gravity anomaly. Thus, gravity derivations in Section 4.2 will be solely based on LSC. Furthermore, LSC is able to combine heterogeneous data for gravity anomaly derivation and this function is needed in our experiments. In all cases, the sea surface topography (SST) is regarded as zero, so that SSH is considered identical to geoidal height. In fact, there is no reliable estimate of SST around this region.

We used the standard remove-restore procedure in the LSC derivation of gravity anomaly. Along-track geoid gradient derived from SSH is used as data type. The adopted reference gravity field is a combined gravity field from the GRACE GGM02C model (degrees 2 to 200; GRACE home page) and the EGM96 model (degrees 201 to 360;

Lemoine et al., 1998). First, a residual geoid gradient is computed by

long

res e e

e = − （14）

where , and are observed, long wavelength and residual gradients, respectively. The residual gravity anomaly is computed by the standard LSC formula with geoid gradients as input (Hwang and Parsons, 1995):

e e_long e_res matrices for gravity anomaly-gradient, gradient-gradient and noise of gradient, respectively. is a diagonal matrix holding the noise variances of geoid gradients.

Based on numerous tests, it is found that a noise variance of 4 sec

eres C_∆ge C_ee C_nn

Cnn

2 for geoid gradient produces the best gravity anomaly. In the case of using combined altimeter and airborne gravimeter data, a residual gravity anomaly is computed by

( )

where is a vector of residual airborne gravity anomalies, , , and

are covariance matrices for airborne gravity anomaly-gradient, gradient-gravity anomaly, and gravity anomaly-airborne gravity anomaly, respectively, and is a diagonal matrix holding the noise variances of airborne gravity anomalies. In this paper,

∆gar C_∆gae C_e∆∆g

∆g∆ga

C

Cmm

a noise variance of 9 mgal² is used in (Hwang et al., 2005).The LSC method in eq.

(16) combines downward continuation, altimeter-to-gravity conversion and interpolation in one step using proper covariance functions.All the needed covariance functions and matrices were constructed based on the error degree variances of the combined GGM02C-EGM96 gravity field and the Tscherning and Rapp (1974) Model 4 degree variance; see Hwang and Parsons (1995, Appendix A) for the detail of covariance modeling related to this work.

Cmm