Echo samples and statistics

The synthetic altimeter echoes are generated over the known three surfaces in fig.5.1. A description of how the synthetic echo is calculated is described in §2.2.6. When the altimeter echoes are generated, 12ns is used for the temporal sampling interval of the altimeter echo, and a pulse duration equivalent to 1.8m in height. This value is selected based on the ice mode operation of the ERS-1 satellite altimeter. A spatial sampling interval for the echoes over surface (a) of 800m is used, and 350m is used for surfaces (b) and (c). These intervals are larger than the realistic interval and they are selected because calculating too many echoes will consume a large amount of computing resources. The tracks are regularly-spaced and parallel to each other with a track spacing which amounts to the sampling interval of the echoes. As mentioned in §3.5.3, the assumption of regular-track spacing can reduce the computational requirement because of the spatial shift invariant property. In consequence, only a single across-track filter in the separable estimation of the multi-channel processing method is needed to calculate each surface retrieval.

After the echoes are generated, the on-board filtering is taken into account. Firstly, speckle noise is added to each echo, 50 different times, and they are averaged together to form a telemetered echo with N=50 (2.42, 2.44). Although in practice the echoes are spatially averaged, from which some of the surface information may be lost and/or a blurring of the echo may occur, for N=50, the increase of the bias due to this effect is insignificant relative to the improvement of height bias due to the decrease of speckle-noise during the filtering. This point has been verified in fig.4.8 - fig.4.10. When N is less 100, the loss of surface information is too small to have an effect on the accuracy of the measurement.

The altimeter echoes over the undulating surfaces are irregular and non-unique in their profiles. The rougher the undulating pattern is, the more unpredictable the features is of the echo profile. This relation can be easily verified by examining some typical echo waveforms in fig.5.4 returned from the three different surfaces shown in fig.5.1. The straight line drawn on the range window indicates the real surface height at the altimeter nadir point. Echoes (a) are returned from surface (a). This surface has a

correlation length 25km, the same as the diam eter of the altim eter footprint. The m ajority of the echoes from this surface are ocean-like and the leading edge of the waveform is not far away from the location of the real surface height.

When the surface becomes rougher, some other profiles emerge. Echoes (b) in fig.5.4 show two typical waveforms returned from surface (b) of 8km correlation length. The first echo on the left hand side is described as a "double-ram p" profile (fig.2.8b) and the right hand side is ocean-like (fig.2.8a). On average, less than half of the echoes from this surface have a surface height lying near either the first-ramp or second ramp of the echoes. In case (b) the left-hand echo has the surface height near the second-ramp, but the echo on the right-hand side has the surface height about 15m further away from the leading edge.

I 1 I i 0 1 0 .5 0 1 0 .5 0 - 5 0 0 R a n g e in m etre 5 0 K - 5 0 0 50 - 5 0 0 5 0 1 t3 0.5 0 - 5 0 1 C 0 .5 0 50 - 5 0 50

Fig.5.4 Samples of altimeter echoes with speckle-noise level N=50. In the figure the straight line indicates the true surface height, and the delay-tim e has been converted into range. Echoes (a) are returned from surface (a); echoes (b) from surface (b); echoes (c) from surface (c). These surfaces are shown in fig.5.1.

Surface (c) of correlation length 4km shown in fig.5.1 has the roughest undulating features. The waveforms of the altimeter echoes returned from this surface are very irregular as shown in echoes (c) in fig.5.4. They fail to fall into any category

of waveform models described by Martin et al. (1983) in fig.2.8. When we examine the echo waveforms in fig.5.4, and consider the assumptions about the echo profile and its relation to the surface height made in the retracking method, we anticipate that the bias will be higher in retrieving surfaces (b) and (c) than in retrieving surface (a)

The ensem ble means of the altim eter echo (2.60) have been exam ined in fig.2.4. It is dependent on the vertical scale but not the horizontal scale of variation. Fig.5.5 com pares the ensemble means of the sampled echoes from three different surfaces in fig.5.1 with their theoretical values. These surfaces have the same vertical but different horizontal scales of undulation. Although the echo waveforms for each surface are very different to each other as shown in fig.5.4, their sample means are about the same and agree with the theoretical ensemble (2.60). The differences shown in fig.5.5 are considered as a statistical effect which can be reduced if the sample size is increased. 0.6 Ê 0 . 4 0.2 - 3 0 -2 0 - 1 0 R an ge w indow

Fig.5.5 Ensem ble mean of altim eter echoes. In the figure the solid curve is the theoretical ensemble mean calculated from (2.60). The curve indicated by the crossed symbols, "x", is the ensemble mean of altimeter echoes returned from surface(a); the curve indicated by the circled symbols, "o", is from surface (b); the curve indicated by the plus symbols, is from surface (c). These surfaces are shown in fig.5.1.

Unlike the ensemble mean of the altimeter echo, the echo variance (3.22, 3.15) is dependent on the surface correlation length. This effect has been exam ined in fig.3.1. In fig.5.6 the variance of the sampled altimeter echoes from the three surfaces in fig.5.1 is compared with their theoretical values. The theoretical values are indicated by the dotted line and the sample variances are indicated by the solid line. Curves (a) are the variances of surface correlation length 25km . The num ber of echoes determining this sample variance is about 44,500. Good agreement is seen between the samples and theoretical values in curves (a). Curves (b) and (c) in fig.5.6 are the

g o 08 > 0 . 0 6 0 . 0 4 0.02 j..L4-4.4. + + + + + + + + + + + + t + - 3 0 -2 0 - 1 0 0 10 20 3 0 R a n g e bins

Fig.5.6 Variance of altim eter echoes. In the figure the dotted curves are the echo variance function calculated from (3.22, 3.15) with param eter Og=20m and the solid curves are the sample variance of the altimeter echoes returned from the three surfaces in fig.5.1. The theoretical variance shown in curves (a) has param eter L=25km and the sample variance is calculated from the echoes returned from surface (a); the theoretical variance shown in curves (b) has param eter L=8km and the sample variance is calculated from the echoes returned from surface (b); the theoretical variance shown in curves (c) has param eter L=4km and the sample variance is calculated from the echoes returned from surface (c).

echo variances from the surfaces of correlation length 8km and 5km respectively. The number of echoes determining these sample variances are 13,600, which is a few times less than the sample size of curves (a). Thus, the agreement of curves (b) and (c) are worse than for curves (a). Although curves (b) and (c) in fig.5.6 are not in good agreement with their theoretical values, the sampled variances are consistent with the proportional relation of the echo variance and the correlation length. We believe that when the sample size of echoes in surface (b) and (c) increases, the difference between their sample and theoretical variances will decrease.