1.1. Situation

For the first time, the ERS-1 altimeter provides observations at high-latitude regions (82°). Figure 1shows the topographic map of the Dome C region from ERS-1 data. Our analyzed satellite track and the region of SPOT coverage are superimposed. We used the WAP (waveform altimeter product) data of the 35 d repeat orbit delivered by the UK-PAF (United Kingdom Processing and Archiving Facility).

The altimetric height is deduced from a retracking technique (fitting process), as explained in Reference Féménias, Rémy, Raizonville and MinisterFéménias and others (1993)and Reference LegrésyLegrésy (1995). The precise orbit from the Deutsch Processing and Archiving Facilities is used. In order to reduce the residual height errors (retracking, orbit and propagation of the radar wave in the atmosphere), we used a total inversion technique (see Reference Rémy, Mazzega, Ноurу, Brossier and MinsterRémy and others, 1989; Reference Brisset and RémyBrissct and Rémy, 1996) in which the signal and errors are separated using their covariance properties. The a posteriori precision of the topography is within 1m.

1.2. Waveform description

The retracking technique allows us to estimate four parameters, as shown in Figure 2:

Fig. 2.

Waveform parameters description, σ° is the backscattering coefficient related to the waveform integral (in dB). Tr is the half leading width, expressed in gates. Fl is the trailing-edge slope in a logarithmic scale (in Np/gate). t₀is the altimetric impact measurement expressed in gates (1gate = 3125 ns or 47 cm in height). The corresponding kilometer scale of the temporal footprint evolution is also given.

The total energy return, related to the backscattering coefficient σ°(expressed in dB). If calculaled on the whole waveform, this represents the energy received from the footprint (e.g. a 5–10 km scale; see Fig. 2).The total energy returned is corrected for the shift of the waveform inside the temporal-acquisition window due to tracking error (Reference Rémy, Brossier and MinsterRémy and others, 1990).

The half leading-edge width, named Tr (expressed in altimetric gates, e.g. 47 cm for ocean mode), which is obtained by fitting the error function (erf). This corresponds to the first part of the impact of the radar wave on the ground surface (e.g. 1 km scale).

The trailing-edge slope, named Fl (expressed in 10⁻Np/ gate), which is obtained by a linear regression of the end of the waveform on a logarithmic scale (from the maximum amplitude gate to the 63rd gate). This corresponds to whole altimeter footprint scale.

Two different heights are extracted: the first one is given by the position of the middle of the leading edge, and the second one by the position of the first echo (by adding the half leading-edge width to the first height).

1.3. ERS-1 data profile

The height profile along track 5056 (marked in Figure 1) is displayed in Figure 3. Panel a shows the height profile, superimposed on the two topography residuals obtained by subtracting a large-scale polynomial form); panels b-d show the three waveform parameters, σ°, Tr and Fl respectively. The two residual topographies in Figure 3acorrespond to the two different heights defined before. The “first impact” topography has larger amplitude than the “first echo”, and is noisier. In general, looking for the middle of the leading edge smooths the topography at the impact footprint scale, so some small-scale features may disappear. Note, however, that some small features which were not visible on the “first echo” topography remain on this topography (see km 130 or 205). The rms of the topographic residuals is 1m. Despite this very smooth topography, the three waveform parameters shown in Figure 3b-d are highly noisy at a 10 km scale. As suggested by the comparison between the three profiles, most of the medium-scale features vary together: see km 50, 70, 85, where local maxima in σ°correspond to minima in Fl and Tr.

Fig. 3.

Different parameters’ profiles along the track: (a) the height profile and two types of residues, one corresponding to the half-leading-edge impact (solid line), the other corresponding to the “first impact echo” (dashed line); (b) the backscaterring coefficient profile; (с) the leading-edge parameter; (d) the trailing-edge slope expressed in 10⁻¹Np/gate, all with local mean values (dashed).

However, this remark does not apply to other locations, for example, the beginning of the profile, where there is higher noise level. A more detailed analysis can be made by comparing real and simulated profiles, as in section 2.4.

The average value of Tr is 3.7 gates, with a rms of 1.9 gates. The average value of Fl is −63.5 × 10 ⁻⁴Np/gate, and the rms is 108 × ⁻⁴Np/gate.

A large-scale signal is also observed in Figure 3; the mean leading-edge value for the first 80km is 1–1.5 gates greater than that for the end of the track, while σ°shows a large-scale trend, with an increase of 1.5 dB from the beginning to the end of the track. The rms of the residual signal is 0.5 dB. Note that the trailing edge shows no significant large-scale variation.

1.4. Mechanisms acting on waveform shape

At the scales studied here, most of the waveform shape and intensity variations are probably caused by small- and medium-scale topographic signals. Several mechanisms may affect the radar echo.

First, the radar waveform shape is very sensitive to the local Surface slope, because of the slope error: the impact of the radar wave does not occur at the nadir where the antenna gain is maximal, but is shifted to an upslope point, nearest to the satellite, where the energy emitted and then received by the antenna is diminished. If the surface slope is α, the impacl occurs at a distance hα(Reference Brenner, Bindschadler, Thomas and ZwallyBrenner and others, 1983: Reference Rémy, Mazzega, Ноurу, Brossier and MinsterRémy and others, 1989), where his the satellite height, and the energy received from the surface will be diminished by a factor exp(−Gsin²α) (Reference BrownBrown, 1977) where Gis the antenna gain (G= 11280 for ERS-1). This will then affect the shape and intensity of the waveform.

Secondly, waveform shape is also very sensitive to the curvature of the surface due to the presence of undulations because of the change in the temporal evolution of the surface illuminated by each individual echo. The signal is focused when the impact occurs in a positive (concave) curvature, while it is dispersed on a negative (convex) curvature. This effect appears at the same wavelengths for both the topography and the waveform parameters.

Finally, there is also evidence of the presence of a volume echo signal. The mean value of the trailing-edge parameter is higher, and the leading edge is lengthened, in comparison to the reference values (−121 × 10⁻⁴Np/gate for Fl, including earth curvature, and 1–2 gates for Tr) in this smooth region where sastrugis are reported to be 30 cm high (Reference VincentVincent, 1994).

2.1. Altimeter simulation

We used a waveform simulation to investigate the phenomena acting on the altimetric signal and to test their effect on the waveform shape.

The waveform simulator is a simplified version of the one developed by Reference Féménias, Rémy, Raizonville and MinisterFéménias and others (1993); the surface is digitized on a 15 m × 15 m regular grid, larger than the footprint of the whole waveform in order to avoid ambiguity at the boundary conditions (in the presence of slope, the footprint is displaced upslope of the nadir and it is expanded in the presence of macroroughness). The time of arrival of the backscattered signal from each illuminated point is calculated along with the distance of this point to the satellite, taking into account its position inside the footprint and local topography. Earth curvature, large-scale topography, undulations or sastrugis can thus be taken into account.

Each point contribution is weighted by the antenna gain pattern or local incidence angle. It is also possible to simulate volume echo, by adding attenuated echoes, shifted in time.

2.2. Topography effects

In order to model the effect of the actual topography on the backscattered signal, we introduced the observed along-track altimetric profile, while the across-track direction topographic signal was extrapolated from the topographic map. Because of the slope error, the profile as seen by the satellite is not the true one. Three methods of correcting for this effect are found in the literature. It is possible to correct the altitude at the nadir, by removing hα²/2from the topography seen by the altimeter (the so-called direct method; Reference Brenner, Bindschadler, Thomas and ZwallyBrenner and others, 1983) or to correct both the position öf the impact point (shifted of hαand the height (by adding hα²/2,the relocation method; Reference Brenner, Bindschadler, Thomas and ZwallyBrenner and others, 1983). One can also demonstrate that between the nadir and impact points, an intermediate point exists where the measured height is the true one (intermediate method; Reference Rémy, Mazzega, Ноurу, Brossier and MinsterRémy and others, 1989). A good recovery of the leading edge needs a very precise modeling of the interaction between radar wave and topography at the impact (Reference Féménias, Rémy, Raizonville and MinisterFéménias and others, 1993). As a consequence, the best method of Correcting for slope error, in the present case, is the relocation method, because it conserves the small-wavelength topographic features near the impact point. Finally, we used a linear interpolation function in order to get correct position and height on a regularly spaced grid (as suggested by Reference Brenner, Bindschadler, Thomas and ZwallyBrenner and others, 1983). Waveform simulation is also affected by slope error, so we can directly compare the observed altimetric profiles and the simulated one.

Fig. 6.

Simulation: the satellite track crosses a band of enhanced backscattering. The three simulated waveform parameter profiles are represented.

2.3. Surface and subsurface signals

2.3.1. Geophsical parameters estimation

The radar echo is constituted by reflection from the air/snow interface and also by internal reflection from subsurface layer interfaces. Inside the snow pack, the subsurface signal can be attenuated by losses by absorption, by scattering by ice grains, and by multiple reflections inside the snowpack. The subsurface signal consists either in scattering by ice grains (Reference Ridley, Cudlip, McIntyre and RapleyRidley and Partington, 1988) or in internal reflection by the subsurface snow layering due to density variations (Reference Rott, Sturm and MillerRott and others, 1993; Reference Rémy, Féménias, Ledroit and MinsterRémy and others, 1995). Using the Seasat scatterometer data above Antarctica, Reference Ledroit, Rémy and MinsterLedroit and others (1993)showed that the contribution of volume scattering by ice grains to the re-emitted signal can be neglected compared to surface and internal interface backscattering when the incidence angle is less than 10°, in the case of altimetry (note, however, that volume scattering by ice grains contributes to the extinction). Moreover, Reference Rémy, Féménias, Ledroit and MinsterRémy and others 1995showed that the internal layering effect is the dominant subsurface signal for any incidence angle in the case of altimetry.

Reference Jezek and AlleyJezek and Alley (1988)reported internal density variations of the order of 0.05 gcm⁻³each 5 cm at Dome C. They also reported penetration depth of the radar waves in the snowpack of 5–10cm for the present frequency (see also Reference Ulaby, Moore and FungUlaby and others, 1986), leading to an extinction coefficient of 0.1–0.2 m⁻¹. Thus, the losses by absorption or ice-grain scattering and the internal interface backscatter must be taken into account in our analysis.

At the surface, it has been demonstrated that the variations of the surface backscattering coefficient on a rough surface can be expressed (Reference Fung and EomFung and Eom, 1982) by:

(1)

where R_s²is the Fresnel coefficient at the air/snow interface and f(s)is linked to the surface slope distribution.

The Fresnel coefficient R²at an interface between two differnt media is given by:

(2)

The snow dielectric constant ∈depends only on the snowpack density ρ(Reference Tiuri, Sihvola, Nyfors and HallikainenTiuri and others, 1984):

(3)

This yields to ∈= 1.68 for a snow density of 0.35 gcm⁻³, and ∈= 1.792 for a snow density of 0.4 gcm ⁻³. The surface Fresnel coefficient is then R_s²= 0.0166 (∈₁= 1, ∈₂= 1.68), leading to a σ_sof 0.0166f(s).

The subsurface signal may be modeled as the contribution of each layer shifted in time and attenuated by both absorption and scattering by ice grains. The reflections occur at each interface and can also be modeled by Equation (1)where R_s²should be replaced by R_int²corresponding to the internal density variations.The internal reflection coefficient is then R_int²= 2.567 × 10 ⁻⁴each 5cm. Assuming that internal layers have the same mirroroughness characteristics as the surface (see Reference AlleyAlley, 1988), each layer (i)will contribute as:

(4)

Here, I₀is the incident flux. k_vis linked to losses by absorption and scattering by ice grains, x_e, is the extinction coefficient linked to all listed effects (2(k_v+ R_int²)), and δ₁= 5 cm.

Each altimetric gate corresponds to с’τ= 30 cm inside the snowpack τ =3 ns, and inside a snowpack of density 0.35 gcm⁻³, the light speed is c’ = ncwhere nis the refraction index inside the snowpack), covering six of the observed layers. The integration gives a λ_jcontribution for layer (j)of δ₂= 30 cm thickness:

with

(5)

where σ_vis the equivalent backscattering of a 30 cm layer. σ_vis then 0.00147f(s)for each 30cm layer. σ_v/σ_sis then equal to 0.09, i.e. mostly sensitive to internal density variations. Note that the contribution of extinction is found to be negligible in the integration.

Values of σ_v/σ_s~ 0.1 and x_e~ 0.1 to 0.2 m⁻¹seem to be very sensitive to surface and subsurface snow properties, principally to the number of internal layers per meter. However, these values remain representative of a realistic case and will be used hereafter.

2.3.2. Computation

One can also take account of the scattering distribution function, due to surface macroroughness, such as sastrugis (named pdf by Reference BrownBrown (1977)), by convolution of the smooth surface response P₂and of the pdf given by:

(6)

where P_sis the surface response to the incident radar impulse, nis the limit of definition of the pdf. σ_sis the surface backscattering coefficient, kis the arrival gate, and Pis the flat-surface impulse response.

The subsurface signal can be seen as the addition of shifted and attenuated echoes (Reference Jezek and AlleyJezek and Alley, 1988). The total energy emitted by snowpack heterogeneities is proportional to the illuminated volume, which is proportional to the illuminated surface (the maximum incidence angle between the radar wave and the surface is very small, ~1°). The received energy is also weighted by the antenna gain which acts similarly on the surface part or volume part, file calculation scheme is then:

(7)

where P_vis the volume signal, x_eis the extinction coefficient of the medium, and σ_v/σ_sis the proportion of energy either scattered or reflected by one layer of 0.3 m thickness. This calculation scheme corresponds to a convolution. In fact, the whole signal can be seen as a convolution between the surface echo and a volume function as:

(8)

2.4. Simulation along a track

The three simulated waveform parameters are superimposed on the observed ones in Figure 4, for the track shown on Figure 1.

Fig. 4.

The simulated parameters, taking account of topographic effects and volume echo (σ_v/σ_s= 0.1 and x_e= 0.15 m⁻¹(dashed line), and Note that for σ° the simulation reproduces most of the peaks, but not the large-scale evolution. The leading edge is very well reproduced; there is also a large-scale signal not reproduced. The trailing-edge variations are not well reproduced in either amplitude or position.

The simulated leading-edge variations are in very good agreement, in both phase and amplitude, with the observed one (see km 10, 90, 110–140, 160, to the end of the profile). However, the ~50km scale amplitude variations of this parameter are not reproduced in the simulation.

The backscattering coefficient is in good agreement, except for the large-scale signal at the end of the track, which is clearly not due to topography features. Most of the observed variations are effectively reproduced (e.g. km 150). However, at the beginning of the profile, the simulation does not restore the whole amplitude on a 20 km distance scale.

The trailing-edge slope is not really in good agreement. Only a few peaks are well scaled (km 30 or 160), some are out of phase (km 210) and most have a diminished amplitude.

There are several reasons for the poor recovery of both the σ°and Fl amplitude variations. First, the across-track signal is poorly known, and thus information is missing for the simulation. Furthermore, the backscattering coefficient depends on the incidence angle. This dependence is very difficult to model because it is sensitive to the microroughness distribution (Reference Ulaby, Moore and FungUlaby and others, 1986) which is not known. This microroughness effect on σ°and Fl enhances the surface curvature effect (the average incidence angle is diminished in a hollow where σ°is maximum and is conversely increased on a bump where σ°is small). Finally, snow properties may exhibit small-scale variations linked with undulations: for instance, Reference Seko, Furukawa, Nishio and WatanabeSeko and others (1993)showed that the accumulation rate is greater in hollows than in bumps. Ihe trailing-edge calculation is also highly noisy because the waveforms are noisy. As a reference, we computed a 300 km ocean track and found a mean of −124 × 10⁻⁴Np/gate with a rms of 95 × 10⁻⁴Np/gate, which is similar to the present observations above the ice sheet.

3.1. SPOT images

The four consecutive SPOT images of this area (Fig. 5) show heterogeneities on the radar footprint scale, such as sastrugis, and medium- to large-scale apparent reflectance changes. The radar footprint is shown on the SPOT images at the corresponding scale. The altimeter track is superimposed with the distance scale.

Fig. 5.

SPOT image of the region, constructed with four consecutive images. The ERS-1 studied track is represented with a kilometric scale. The footprint of the altimeter radar wave is represented in the lower left corner; the central cirle corresponds to the first impact. The apparent reflectance of the surface is coded in 23.NC (number of counts). They revealed the presence of small-scale heterogeneities. The large-scale signal is principally caused by spatial variations of sun elevation. One can see sastrugi fields at the top, clear spots spread across the images, and strong albedo-change area. The central detail is a cloud.

We used the following system of equations to interpret the SPOT images:

The intensity of the apparent reflection is coded on a linear scale of numerical counts (NC), linearly linked with the irradianсе of the satellite (L_sat) which is given by:

(9)

where θ_iis the solar incidence angle on the surface, ρ_appis the apparent reflectance of the surface, E₀is the incident solar radiation at the top of the atmosphere, and Dthe losses throughout the atmosphere.

The observed variations are then relative (Reference Bourdelles and FilyBourdelles and Fily, 1993), and are given by:

(10)

Variations of observed NC are then due either to local incidence angle or to albedo changes. One NC variation yields a 0.5% change in albedo at 192 NC, which is the mean value of the images (values vary from 182 to 203), while a 1 NC variation yields a 0.67° change in local slope (or incidence angle) at 24°, which is approximately the sun elevation at the snapshot time.

Note that the images are not corrected for the spatial variation of sun elevation throughout the images, which is the principal signal (10 NC). Nor are they corrected for problems in the data that make four stripes in four sections. However, these processed SPOT images reveal three different types of observed structures that can be analyzed in relation to our altimetric signal.

3.2. Roughness

One can observe fields of sastrugis which are irregularly spread, particularly at the top of the image (km 25–80). Assuming a constant reflection coefficient and a “factory roof” structure, we calculated that they correspond to a slope of 1% on 10 pixels. This is equivalent to a structure 1 m high, 100 m wide and about 2 km long. These structures, which are not visible in situ, have already been observed by aerial observations (Reference Bromwich, Parish and ZormanBromwich and others, 1990).

The altimetric waveform leading-edge parameter shows great variability and a 1 gate increase in this region (see km 20–80, especially in the first 10 km, around km 40 and km 60–80 (Figs 3c and 5)), which effectively corresponds to the mean increase due to sastrugi fields. It also corresponds to the small values of σ°poorly recovered by the simulation. Finally, these sastrugi fields probably enhance the dependence of the backscattering on the incidence angle, and may explain the great amplitude of the σ°and Fl variations at the beginning of the profile.

3.3. Changes in brightness

Clear spots of strong brightness changes are also observed on SPOT images. For instance, a very clear albedo change occurs from km 175 to the end of the profile, and albedo changes of а few kilometers occur near km 160. Looking at Figures 3b and 5, one can observe that the enhancement of altimetric backscattering coefficient clearly corresponds to the large-scale change in brightness. Because of the radar footprint scale, the abrupt change is smoothed over 10–20 km (see Fig. 2). The other two waveform-shape parameters, Tr and Fl, are not clearly sensitive to these structures. These clear spots seem to be related to microroughness changes, because, first, only microroughness variations may explain such local albedo changes (Reference Choudhury and ChangChoudhury and Chang, 1981; Reference Bourdelles and FilyBourdelles and Fily, 1993), and, secondly, this signal affects only σ°, without changing the other waveform parameters. Only roughness changes may affect surface + volume signal without affecting σ_v/σ_s(Equation (5)). This is Confirmed by in situ measurements, which snow that these spots do not correspond to hard changes in snow properties but effectively correspond to abrupt changes in surface roughness personal communication from F. Candaudap and M. Fily, 1996).

3.4. Clear bands

Finally, we observed clear bands tens of km long and several km wide. The variations of the apparent reflectance (4–5 NC) yield slope Variations of ~ 1.2%, which are not in accordance with any surface feature seen by the altimeter.

This factor 10 of discordance in slope estimation cannot be due to error either in altimetric measurement or in Equation (9). A possible explanation is that, at this time of year (November), the elevation of the sun is so low that surface characteristics such as temperature are very sensitive to surface slope variations of a few percent personal communication from P. Reference PetréPetré, 1995).

It is also possible that some features in Figure 5result from obstruction by thin cirrus clouds and cloud shadows, but we have no independent measurements to confirm this hypothesis.

Another explanation is that real snow optical properties (snow grain-size, snowpack density or surface roughness: dρ/ρ~2.5%) are effectively related to undulation features, as demonstrated by Reference Seko, Furukawa, Nishio and WatanabeSeko and others (1993).

4.1. Local change in backscattering coefficient, and effect on altimetric parameters

Observations suggest that sastrugi fields, or abrupt microroughness variations, or perhaps snow-property variations, create local backscattering variations. How do these backscattering variations affect the waveform shape?

We simulated the crossing of a band of different backscattering by the altimeter. The band is 2 km wide, the backscattering is enhanced by 1 dB, and the angle between satellite track and the band is 45° (see Fig. 6a).

As suggested by Figure 2, the last gates of the waveform are sensitive to the band 5 km before the intersection between the satellite track and the band. The trailing-edge slope then begins to rise until reaching −90 × 10 ⁻⁴Np/gate, 5km before the intersection, and decreases to −150 × 10 ⁻⁴Np/gate in the middle of the band (Fig. 6). The leading edge acts similarly but at a smaller scale with a smaller amplitude (0.25 gate). The σ°is of course enhanced, but only by a factor of 0.3 dB, corresponding to the weighting between both surface types in the footprint.

The effect is greater on the waveform shape than on the total received energy. This may partly explain the strong variations in Fl values. The height measurement is then only affected by a few centimeters. Reference Ridley, Cudlip, McIntyre and RapleyRidley and others (1989)have already shown that the crossing of a ridge in West Antarctica affects the altimetric waveform, but it is the first time that such effects have been found to play a role in the smooth interior of the continent.

4.2. Discussion on subsurface sensitivity

Figure 7shows the altimetric parameters’ evolution along the track shown in Figure 3, with respect to different volume configurations (according to section 2.3, σ_v/σ_s= 0.1, x_e= 0.2, 0.15, 0.1). The effect of the subsurface signal is not the same for different parameters. Only the mean value of σ°or Fl is affected; no fluctuation due to the initial surface echo shape is visible. Thus, the presence of a small spatial-scale signal will not compromise the retrieval of geophysical quantities using σ°or F1.

Fig. 7.

Simulated parameters for different cases. The surface signal is drawn in solid line, and the three cases where the subsurface features are added to surface signal are drawn in dashed. The σ_v/σ_scoefficient is 0.1 for the three cases and the extinction varies from 0.2 to 0.15 down to 0.1. Note that the variations in σ° and trailing edge linearly depend on the volume signal, but the leading-edge changes are more complicated.

On the other hand, the subsurface effect on the leading edge is strongly dependent on the surface echo shape and introduces a kilometric-scale signal. The addition of the subsurface part induces a complex effect. For instance, at km 50 or 70, the leading edge is little affected by the variation of the subsurface signal, while at km 150 or around 200, the leading edge is enhanced by more than 2 gates, and produces fluctuations which were not present before. Except at the beginning of the profile, where the large spatial-scale feature is not recovered, the whole signal is then due to a complex combination between topography and penetration effects. Note that varying σ_vproduces comparable effects. This means that even in this smooth area, few gates of variations in the leading edge can result from these complex effects and have direct small-distance-scale impact on the height estimation. This is confirmed by analysis of the dual-frequency Topex altimeter above Greenland: the height difference between the C and Ku bands shows a strong small-scale signal (Reference Rémy, Legrésy, Bleuzen, Vincent and MinsterRémy and others, in press). As a consequence of this complex effect, the retrieval of surface and subsurface parameters should not be performed with the help of the beginning of the waveform, as is currently done (Reference Davis and MooreDavis and Moore, 1993; Reference Yi and BentleyYi and Bentley, 1994). For instance, Reference Féménias, Rémy, Raizonville and MinisterFéménias and others (1993)have shown that it is possible to have two similar wave forms with two different volume configurations. Alternatively, Reference LegrésyLegrésy (1995)shows that it is possible to have very different waveforms with the same volume characteristics, depending on local topography even by averaging many waveforms .

The combination between surface geometry and subsurface effect depends on the altimeter characteristics. We simulated the profile shown in Figure 3, as seen by both the Seasat and ERS-1 altimeters. The volume signal is characterized by x_e= 0.15m⁻¹and σ_v/σ_s= 0.1. Even with the weak differences between the two missions (antenna width of 1m for Seasat and 1.2 m for ERS-1, and altitudes of 800 km for Seasat and 780 km for ERS-1), an average height difference of 6.3 cm with a rms of 11.4cm is found (Fig. 8). This effect is similar if other retracking data such as WAP ones are used (Reference LegrésyLegrésy, 1995). The effect is small in this case, but particular care should be taken for other altimetric missions and other regions.

Fig. 8.

Height difference of the same profile seen by Seasat and ERS-1 altimeters. The difference between the two satellites is 20 km in altitude and 20 cm in antenna diameter. The simulation is performed wilh a volume signal characterized by σ_v//σ_s= 0.1 and x_e= 0.15 m⁻¹. In this smooth region, the simulated difference reaches 6.3 cm with a rms of 11.4 cm.

In summary, therefore, at a 50 km scale, the averaged σ°or Fl values seem to be characteristic of the geophysical processes, whereas the leading-edge signal shows a non-linear response which is more difficult to interpret.