Best linear estimation of multi-channel processing.

In theory, best linear estimation technique is able to resolve the along-track and across-track topography by processing the altimeter echoes in both track directions (3.1). However, it is practically impossible as discussed in §3.4. Although the separable estimation is basically practical, it is an extremely time-consuming process when it is applied in practice. In our calculations, the across-track spacing is assumed regular in order to simplify the calculation of the across-track filters in the separable estimation because of the spatial shift invariant property, as described in §3.5.3. Realistically, this is not the case, because the ground tracks are irregularly spaced, and in consequence the calculation of the across-track filters will also be very time- consuming because the spatial shift invariant property does not apply.

Unlike the across-track echoes, the along-track echoes are regularly sampled by the altimeter, and the spatial shift invariant property can be applied to the estimator. If only the along-track echoes are considered in the best linear estimation, the calculation will be very much simplified, which is more practically suitable in a realistic situation. However, with the absence of the across-track information in the estimation, one can anticipate its performance will be worse than the performance of the separable estimation but better than the retracking method, as the topography in one direction will be resolved in the best linear estimation.

6.2.1 Quantitative assessment.

The RMS errors of the retrieved surfaces obtained from the best linear estimation (3.1) are summarised in table.6.2. In the estimation, 30 altimeter echoes in an along-track are used. The first column is the RMS error of the retrieved surfaces and the second column is the a posteriori error (3.6). The a posteriori errors are

RMS error z posted ori error Surface(a) Gs=20m, L=25km 1.65m 2.63m Surface(b) as=20m, L=8km 4.65m 5.81m Surface(c) as=20m, L=4km 8.24m 9.04m

Table.6.2 The RMS error of the retrieved surfaces obtained from the best linear estimation. The first column shows the RMS error obtained from 30 along- track echoes in the estimation and the second column shows the a posteriori error of the estimate (3.6).

about Im higher than the RMS error of the retrieved surface which is less than one range bin equivalent. We suspect that this minor difference is due to numerical error, especially in the inversion of the covariance matrix where a value is added to the diagonal of the matrix in order to overcome the singularity, as described in §3.5.2. In addition, the generated surfaces are only an approximation of the Gaussian surfaces on which the calculation of a posteriori error is based. From table.6.2, it can be

confirmed that the a posteriori error is a good indicator of the accuracy of the retrieved surfaces. In practice, the true surface is not known a priori, so, in turn, the accuracy of the recovered surface from the retracking method cannot be determined. One of the advantages of the multi-channel processing method is that it allows us to calculate the a posteriori error from which the accuracy of the retrieved surface can be estimated.

By comparing the RMS error from the best linear estimation in table.6.2 with the RMS error from the retracking method in table.5.1, it can be seen that significant bias has been removed by the best linear estimation, especially when the surfaces have smaller correlation length.

Due to a lack of across-track information in the best linear estimation, the accuracy of the retrieved surfaces from this estimation is poorer than the accuracy from the separable estimation. This can be easily verified when the RMS error obtained from the best linear estimation in the first column of table.6.2 is compared with the RMS error calculated from the separable estimation in table.6.1.

6.2.2 Qualitative assessment.

For a qualitative assessment, one dimensional surface profile of the retrieved surface from the best linear estimation is presented in flg.6.6. The true surface is from surface (b) of 8km correlation length shown in fig.5.1. The retrieved surface, indicated by the "plus" symbols, is obtained by using 30 along-track echoes in the best linear estimation. When we compare fig.6.6 with fig.5.9, we can easily see the speckled fluctuation and the high bias in the problem region (80-100km) in fig.5.9 has been significantly removed by the estimate from a sequence of along-track echoes. However, without resolving the across-track topography in the best linear estimation, we can obviously see that the bias of the retrieved surface from the best linear estimation in fig.6.6 is higher than that from the separable estimation in fig.6.3.

In fig.4.5 and fig.4.6, the a posteriori errors have shown that when only the along-track data is used in the best linear estimation, the accuracy of the measurement is not very sensitive to the input pattern of the data. This can be further verified in this

20 -2 0 - 3 0 - 4 0 0 20 4 0) 6 0 f A c r o s s-tr a c k location in km 80 100 120

Fig.6.6 Comparison of an one dim ensional profile of the retrieved surface and true surface. The true surface is at the along-track distance 40km of surface (b) with L=8km as shown in fig.5.1. The retrieved surface is obtained from the best linear estimation of the multi-channel processing method in which 30 altim eter echoes of spatial coverage 10km, and each echo of 63 tem poral samples in a single along-track are used.

simulation of the surface retrieval process. The retrieved surface shown in fig.6.6 is obtained from 30 along-track echoes of spatial coverage 10km. Unlike the separable estimation described in §6.1.2, we have found that, for a given data density, reducing half of the input data coverage in the best linear estimation will give approximately the same quality or bias of the measurements. An example is given in fig.6.7 in order to further demonstrate this insensitivity to the input data pattern. Fig.6.7 illustrates the retrieved surface obtained from the best linear estimation, but with a spatial coverage only half of the coverage in fig.6.6. W hen fig.6.7 is com pared with fig.6.6, where w ider spatial coverage of data is used in the estimation, it can be easily seen that the retrieved surfaces from both figures show sim ilar bias patterns in their retrieved surfaces. In the separable estimation, when the spatial coverage of the data changes, the accuracy of the measurement will significantly change too. From these points, we suspect that, in the best linear estimation, the bias is dominated by the across-track

topographies. -20 - 3 0 - 4 0 0 20 40) 6 0 f A c r o ss-tr a c k location in km 80 100 120

Fig.6.7 Comparison of an one dimensional profile of the retrieved surface and true surface. The true surface is at the along-track distance 40km of surface (b) with L=8km as shown in fig.5.1. The retrieved surface is obtained from the best linear estimation of the multi-channel processing method in which 15 altimeter echoes of spatial coverage 5km, and each echo of 63 temporal samples in a single along-track are used.

Another example given in fig.6.8 is the retrieved surface obtained from the same estimation but when across-track echoes are used instead of the along-track echoes. The bias in the problem region 80-100km has been significantly increased, as compared fig.6.8 with fig.6.6. This is because the sequence of echoes used in the estimation is from the across-track direction, and hence the along-track topography in the sub-satellite track shown in fig.5.10 is not resolved in the estimation. In consequence, the measurements over-estimate the valley in that problem region.

When fig.6.6 and fig.6.8 are compared, the bias patterns of both figures are different. Some regions in the figures are worse, and some better than the other. This

implies that when only echoes from one direction are included in the best linear estimation, the bias due to the topography from another direction will be introduced into the retrieved surface. After observing the simulation results from the separable estimation method and best linear estimation, it can be concluded that, to achieve a higher accuracy of the measurement, the input echoes from both perpendicular track directions are necessary for the best linear estimation (3.1). However, this is a very expensive process which far exceeds our computing capability.

40 £ -C O) I 0 (D Ü ■g -1 0 3 CO -20 - 3 0 - 4 0 120 4 0 6 0 8 0 100 20 0 A c r o ss-tr a c k location in km

Fig.6.8 Comparison of an one dimensional profile of the retrieved surface and true surface. The true surface is at the along-track distance 40km of surface (b) with L=8km as shown in fig.5.1. The retrieved surface is obtained from the best linear estimation of the multi-channel processing method in which 30 spatial and 63 temporal samples in a single across-track are used.