Roughness parameters behaved differently depending on the tillage class. Figure 3 represents the boxplots of ground measured s and l values.

It can be observed that s took larger values as the tillage class became rougher. Generally, fields of different classes had different s values, although the variability of s was large in some classes (particularly the rough ones). The behaviour of l was different, demonstrating for most of the classes a very large variability and a similar variation range. For this study, rough classes had a lower variability range than smooth classes.

In Table 2, field scale average values are presented. Focusing on the field scale spatial variability of the parameters, it can be observed that the average coefficients of variation ranged from 16% to 25% for parameter s and from 38% to 94% for parameter l.

Figure 4 depicts the autocorrelation functions (ACF) obtained for fields belonging to the different tillage classes studied. For each field the mean ACF was computed averaging the individual ACFs. Next, the root mean square error (rmse) of the field average ACF was calculated. This rmse value illustrates the spatial variability of the ACFs. The field average ACF was fitted to the exponential and Gaussian functions and the rmse of their fit was computed. The plots show that the variability of the ACFs can be large, especially at higher lags. To avoid the impact of the larger variability at higher lags the calculation of the rmse values was restricted to a lag up to 50 cm.

The plots illustrated in Figure 4 show that the spatial variability of ACFs can be high in both smooth and rough surfaces. In smooth surfaces, the ACF tends to fall quickly at small lags due to high frequency roughness components. But the slope of the function decreases soon leading in some cases to very large l values. For rough surfaces, the decay is generally more homogeneous with a gradual reduction of the slope. This type of ACF represents surfaces that are rough in both high and low frequency domains. In all cases, the field average ACF fits better the exponential function than the Gaussian. The fit to the exponential function is particularly good for rough surfaces. For smooth surfaces, exponential and Gaussian functions do not provide an adequate description of the initial decay and the sudden reduction of the slope of the ACFs.

For each tillage class the average of the field ACF̅ rmse was calculated and the values obtained are represented in Table 3. It can be observed that the variability of the ACFs was similar in the different classes, with values ranging from 0.085 (‘Rolled seedbed’) to 0.130 (‘Seedbed’). In fact, rough and smooth classes yielded similar rmse values, indicating a substantial variability of the ACFs in all cases. Table 3 also gathers the rmse values of the ACF fit to the exponential and Gaussian functions. It can be observed that the fit to the exponential function is better (lower rmse) in all cases. Generally, rough classes fit better the exponential function than the smooth ones, except for the class ‘Rolled seedbed’.

The discrepancies between the experimental and the exponential ACFs illustrated here show that assuming an exponential ACF in the retrieval of SM could be a significant source of errors. However, the quantification of these errors in terms of the modelled σ⁰ or the retrieved SM has never been done before, as far as the authors know. In any case, this quantification is out of the scope of this paper, so the results described in the next section assume an exponential ACF and do not represent the influence of the discrepancies between the experimental ACF and the exponential function.

In summary, the average standard deviation of s and l on seedbed fields was 0.33 cm and 19.24 cm respectively. In the following sections, the influence that this variability exerts on the simulated backscattering coefficient values and on the retrieved SM estimates, is assessed using IEM simulations.

Figure 5 represents the field average s and l values measured and the IEM validity range for both C- and L-bands. It can be observed that most seedbed fields are within the C-band range, whereas most of the fields of rougher classes are not. At L-band the validity range is much wider and covers most of the measured fields. However, for some fields the s/l < 0.4 condition does not hold, causing their exclusion from further analysis.

To evaluate the influence of the field scale roughness variability on σ⁰, the IEM was run using field average roughness parameters first and then the parameters measured on each profile individually were applied. This resulted in field average and individual backscatter values, called σ̄⁰ and σ⁰_i respectively.

The comparison between σ⁰_i and σ̄⁰ (Figures 6 and 7) illustrates the influence of roughness spatial variability on σ⁰. It should be remarked that the considered sensor configuration and soil characteristics were exactly the same for every field. This means that σ̄⁰ (abscissa in Figures 6 and 7) varied only due to the differences in the field average s and l values from field to field. This field average σ̄⁰ showed a variation range between 5 and 10 dB. This result indicates that assuming tillage class average or reference roughness values could imply high inaccuracies in the modelling of σ⁰.

The intra-field spatial variability of roughness caused σ⁰_i values to be very different from their corresponding field average σ⁰ values in some cases. The root mean square error (rmse) values obtained ranged from 1.36 to 2.29 dB. Highest errors were observed at 45° incidence angle in both C- and L-bands and at 15° in C-band.

The same rmse values were obtained for different SM conditions, so the influence of roughness spatial variability seems not to be affected by the moisture conditions of soils. Very similar errors were observed also for both VV and HH polarizations. The rmse was slightly higher on VV for 45° incidence angles but this difference did not reach 0.5 dB. In the other cases, both polarizations yielded very similar rmse values.

The incidence angle showed a significant influence on the obtained rmse values. For C-band simulations (Figure 6), rmse was minimum at 30° and increased at both 15° and 45°. In the case of L-band (Figure 7), rmse increased with the incidence angle, and highest errors were observed at 45°.

In order to analyse the influence of the incidence angle in more detail, an additional simulation was conducted varying θ_inc from 10° to 55° with a 5° step and considering average moisture conditions (SM=0.20 cm³cm^-3) for HH and VV polarization and both C- and L-bands. Figure 8 represents the rmse values obtained as a function of θ_inc for the different configurations. The trend was similar in all cases, at low θ_inc values errors were high, as the θ_inc increased rmse first decreased to a minimum and then increased.

Minimum rmse values were observed at 15°-20° in L-band and 25°-30° in C-band. At low θ_inc values errors were significantly higher in C-band, but at medium θ_inc the opposite was evidenced, this is in agreement with previous studies [19], which evidenced higher roughness sensitivity at L-band than at C-band on SIR-C/X SAR observations with 26.4° incidence angle. In our study, large θ_inc values gave rmse values higher than 2.0 dB in all cases, which were slightly higher in VV polarization than in HH.

A similar analysis was performed running the inverse IEM algorithm. This time, field average σ̄⁰ values were used as input along with the roughness parameters of the individual profiles. The inverted SM values should be equal to the initially assumed conditions (0.05, 0.20 or 0.35 cm³cm^-3), so the discrepancies observed (evaluated as rmse values) were considered a consequence of the field scale roughness variability.

Figure 9 shows the rmse values obtained as a function of θ_inc for both bands and different SM conditions. It can be observed that independent from the incidence angle trend (which is similar to the one observed in the forward simulations), rmse values were different depending on the SM conditions. For wet soils, rmse values were higher, especially on L-band and on C-band with large θ_inc. In fact, for wet conditions, retrieved SM values were sometimes even higher than the fixed upper limit (0.60 cm³cm^-3).

If we look more closely to the rmse as a function of SM (Figure 10) we can observe a clear relationship, especially in L-band. Errors were close to 0.05 cm³cm^-3 in the driest case and they increased with moisture, sometimes reaching values around 0.15 cm³cm^-3. This phenomenon could be a result of the reduced sensitivity of σ⁰ to soil moisture for higher SM-values. A given error in σ⁰ produces a certain error on the retrieved SM on dry conditions but this error could increase substantially over wet conditions. Similar results have been reported in the literature [13, 17, 33]. This behaviour seems to be more pronounced at L-band, resulting in higher SM retrieval errors on wet conditions than at C-band.

In summary, the results of this synthetic analysis indicate that the intra-field spatial variability of roughness could severely affect the inversion of SM from SAR data. Unless roughness parameters are very accurately measured, retrieval errors could easily reach 0.10 cm³cm^-3, and even more on wet conditions. Consequently, the measurement or characterization of the roughness parameters for each field needs to be carried out with high detail in order to retrieve useful SM estimates.

The temporal variations of s and l were only evaluated for seedbed fields. Measurements corresponding to five dates were analysed (from 24/10/2004 to 01/03/2005). In Figure 11, the temporal variations of both roughness parameters are illustrated. Variations were computed for each field in reference to the measurements of the first seedbed date, so negative values represent a reduction on the parameter and positive values an increment.

The temporal evolution of the ACF variability (rmse of the field ACF) was also analyzed. Figure 12 depicts the rmse values calculated for each seedbed field ACF on the different dates. It can be observed that the variability of the ACFs was similar in the different dates. For some fields the rmse value decreased whereas for others it increased. Therefore, even if surfaces roughness appeared to smoothen with time the variability of the ACFs did not change significantly.

On the other hand, the fit to the exponential function improved with time. Figure 13 plots the rmse of the fit to the exponential function for the different fields as a function of time. It can be observed that the rmse values obtained decreased in most of the fields. This means that the smoothening caused by precipitation resulted in more exponential-like surfaces, probably due to the reduction of the high frequency roughness components of the surface that caused the steep fall at the beginning of the ACFs illustrated in Figure 4 for smooth classes.

The recorded temporal variations of s and l could cause a certain variation on σ⁰. To assess this variation, the backscatter values calculated considering constant, or initial, roughness parameters (σ⁰₀), were compared to those calculated using s and l values of each date (σ⁰_i). The differences were computed as Δσ⁰=σ⁰₀ - σ⁰_i. The objective was thus to assess whether the roughness dynamics caused significant variations on σ⁰.

Figure 14 shows the Δσ⁰ values computed for both bands at 30° incidence angle, VV polarization and 0.20 cm³cm^-3 as a function of time. Variations were similar to those of the roughness parameters themselves; on the first dates σ⁰ increments and reductions were observed with no clear trend. But the roughness variations on the last date caused a clear reduction in σ⁰, especially in L-band. In these conditions both s reductions and l increments made σ⁰ decrease, so even if roughness temporal variations seemed minor, their impact on σ⁰ could be important.

The backscatter variations were calculated for all different configurations considered but they are not plotted for the sake of conciseness. Instead, the total variations between the first and last dates were calculated for the different fields and their average, minimum and maximum values for the different conditions are represented in table 4. As the same Δσ⁰ values were obtained at different SM conditions (as in section 3.1.2), the numbers in table 4 are not restricted to any SM content.

It can be observed that reductions were more significant at large incidence angles, whereas at 15° increments were observed especially for C-band. This could be a consequence of the smoothening of the surface, that caused a specular-like type scattering behaviour producing higher σ⁰ values for steep incidence angles and lower σ⁰ values for large angles. This effect is further illustrated in figure 15 and it is particularly evident at C-band in fields with strong s reductions and l increments (fields 193 and 208). The backscatter variations are minimum at 25°-30° in C-band and 15°-20° in L-band.

In order to evaluate the influence of temporal variations of roughness on the inverted SM, the IEM was run assuming constant roughness parameters (s and l values corresponding to the first date) and using σ⁰ values obtained for each date. The differences (ΔSM) between the inverted SM values and the initially considered conditions (0.05, 0.20 and 0.35 cm³cm^-3) for the different dates were computed. Figure 16 shows the ΔSM values obtained in both C- and L-bands considering 30° incidence angle, VV polarization and a soil moisture of 0.20 cm³cm^-3. On the first dates variations were quite erratic as in the case of Δσ⁰. Nevertheless, on the last date all the fields showed lower SM values than initially assumed (ΔSM values below zero), especially in L-band. The same analysis was carried out in the different sensor configurations and moisture conditions assumed. Tables 5 and 6 summarize the variations (average, minimum and maximum) in the inverted SM between the first and last date on the different conditions, for both C- and L-bands.

The results obtained suggest that roughness temporal dynamics could cause the inverted SM to be underestimated (negative ASM̅ values) if constant (initial) roughness parameters are considered along the growing season. The underestimations seem more severe if the incidence angle is large and the soil is wet. In fact at low incidence angles, especially in C-band, variations could be positive, causing soil moisture to be overestimated. This effect may be a result of the smoothening of soils that caused the σ⁰ (and hence the inverted SM) to increase at steep incidence angles and decrease at large angles as a consequence of a more specular-like behaviour of the surface.

Variations were in the same range in both C- and L-bands, but they were slightly larger in L-band. Underestimations could reach values higher than 0.10 cm³cm^-3, which is an error value that cannot be ignored from an applications' point of view.