The effect of spatial distance and density of the echoes on the measurement.

In the multi-channel processing method, there is no rule for selecting the pattern of the input echoes in the estimation. However, the amount of data used in the estimation is usually restricted by the existing computer resources. In our case, 30 spatial echoes, each of these containing 63 temporal samples, is regarded as the maximum amount of data in the estimation. For such a limited amount of data, the pattern of the echoes can be desampled using a larger spatial interval. In turn, the estimation will have a lower data density but a wider spatial coverage. In contrast, the echoes can be desampled with a smaller-spatial interval which gives higher input density but shorter spatial coverage. In this section, we shall discuss how this selection between the data density and the spatial coverage affects the accuracy of the measurement.

Fig.4.5 illustrates a posteriori error as a function of the desampling interval of the input echoes. The results are calculated from the best linear estimation (3.4, 3.6) which is based on the input of 30 echoes and each echo contains 63 temporal samples. The a posteriori error is calculated for a surface correlation length L=8km for curve (a) and L=20km for curve (b). They both have the same vertical scale Gg=20m. In the figure, the a posteriori error of curve (a) shows that when the spatial interval of the echoes increases, the error first decreases. However, when the spatial interval further increases beyond the minimum point at about 550m, the error gradually increases but in a very slow manner.

The location of the measurement point of the a posteriori error curves in fig.4.5 is at the centre point of the sequence of the input echoes. It can be seen in the figure that curve (a) has the minimum point at the sampling interval 550m. This gives

one-sided spatial coverage for 30 input echoes of about 8km which is the same as the surface correlation length used in the calculation of this curve. From this point, we can understand that when a single-sided coverage of input echoes is sm aller than the surface correlation length, the spatial distance of the echoes is more important than the data density. This explains why the a posteriori error reduces in the earlier part of the curve shown in fig.4.5. After the spatial coverage of the echoes has exceeded twice the surface correlation length, the error curve is almost flat because the effect of a wider spatial width and lower data density are balanced out.

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Fig.4.5 The a posteriori error as a function of the spatial sampling interval. The error is calculated from the best linear estimation (3.4, 3.6) in which 30 echoes are used. The vertical scale of surface variation Gg=20m. Curve (a) has the surface correlation length L=8km; and curve (b) has the surface correlation length 20km.

The a posteriori error in curve (b) shown in fig.4.5 is almost a straight line. W hen the surface correlation length L=20km, the data density and its spatial distance have an equal and opposite effect on the measurement. In consequence, they are totally balanced out and give a constant error curve.

In fig.4.5, we cannot clearly see the individual effects of the data density and the spatial coverage because when one of these variables changes, the other will change too. We shall give another example to observe the effect of the data density when two fixed spatial coverages of the data are used. We present in fig.4.6 the a p o sterio ri error as a function of the number of echoes used in the estim ation. The error is calculated from the best linear estimation (3.4, 3.6) where the altim eter echoes from the along-track direction are used. Curves (a) and (a') are calculated from the surface correlation L= 8km and curves (b) and (b') are calculated from the correlation length L=20km. All error curves have the same vertical scales, Og=20m. Curve (a)

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Fig.4.6 The a posteriori error as a function of the number of the echoes used in the estimation. The error is calculated from the best linear estim ation (3.4, 3.6) and the vertical scale of surface variation is (jg=20m. Curve (a) and (a') have the surface correlation length L=8km where (a) has the spatial coverage of the echoes of 10km and (a') has coverage of 60km. Curve (b) and (b') have the surface correlation length 20km where curve (b) has the spatial coverage of the echoes of 10km and (b') has coverage of 60km.

and curve (b) have a spatial distance of the echoes of 60km and curve(a') and curve(b') have spatial distance 10km.

Fig.4.6 has shown that when the data density increases further, the error tends to converge to a constant value. The insensitivity of the later part of these curves are beyond our expectation. We suspect that when the across-track data is not included in the estimation, which is the case in fig.4.6, the unresolvable across-track topographies dominate the bias of the measurement. Further increase of the data in the along-track direction does not improve the knowledge given to the estimation of the surface topographies in the across-track direction. We have found that the across-track processing in the separable estimation (3.51) is more sensitive to the change of the input data than the best linear estimation (3.1). This is true because the narrow-beam echoes contain the antenna beam orientation which is very narrow in the along-track direction and the sequence of echoes is selected in the across-track direction, hence increasing the number in this sequence will give more desired knowledge to the estimation.

For siuface correlation length L=8km, the later part of curve(a) and curve(a') shown in fig.4.6 respond very slowly to the increase of the echoes. This is mostly due to the lack of across-track data in the estimate. Other factors are the spatial coverage of curve(a) being less than twice the surface correlation length and the low data density of curve(a'). For the surface of 20km correlation length, the a posteriori error curves (b) and (b') behave slightly differently to the error curves (a) and (a') of 8km surface correlation length. We can see in curve (b), where the spatial coverage is 10km, that the error curve has converged to an asymptotic value while the a posteriori error in curve (b'), with the data spatial coverage of 60km, is still decreasing at a relatively fast rate. They intersect at the point where the data density of 10km spatial distance is 2.5 per km and the density of 60km spatial distance is 0.4 per km. This implies that when the spatial coverage is less than twice the surface correlation, increasing the number of echoes does not effectively improve the accuracy of the measurement.

4.4 The effect of temporal duration and interval of an