On the characterisation of agricultural soil roughness for radar remote sensing studies Malcolm W. J. Davidson, Thuy Le Toan Centre D’Études de la Biosphere (CESBIO) 18, av. Edouard Belin F-31401 Toulouse, France Email:
[email protected] Francesco Mattia ITIS-CNR, Matera, Italy Giuseppe Satalino IESI-CNR, Bari, Italy Terhikki Maninnen VTT, Espoo, Finland Maurice Borgeaud ESA-ESTEC, Noordwijk, the Netherlands
Abstract The surface roughness parameters commonly used as inputs to electromagnetic surface scattering models (SPM, PO, GO and IEM) are the root mean square (RMS) height s and autocorrelation length l. However soil moisture retrieval studies based on these models have yielded inconsistent results, not so much because of the failure of the models themselves, but because of the complexity of natural surfaces and the difficulty in estimating appropriate input roughness parameters. In this paper we address the issue of soil roughness characterisation in the case of agricultural fields having different tillage (roughness) states, by making use of an extensive multi-site database of surface profiles collected using a novel laser profiler capable of recording profiles up to 25 metres long. Using this dataset the range of RMS height and correlation values associated with each agricultural roughness state is estimated, and the dependence of these estimates on profile length investigated. The results show that at spatial scales equivalent to those of the SAR resolution cell agricultural surface roughness characteristics are well described by the superposition of a single-scale process related to the tillage state with a multi-scale random fractal process related to field topography.
Acknowledgements The authors would like to thank George Marty, Davor Dobra, Guido Pasquariello, Anna Maria Castrignano and Michele Rinaldiin for their help in the arduous task of collecting profile data. We would also like to express our thanks to the Mayor of Marestaing, M. Lafont, who helped in the initial selection of test fields as well as the farmers of Marestaing and Matera who made their fields available to us for profile measurements. The work presented in this paper was carried out under ESA/ESTEC contract 12008/96/NL/NB. The construction of the laser profiler was financed by ESA/ESTEC under the same contract.
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I - Introduction The observed sensitivity of the radar backscattering coefficient σ0 to soil moisture conditions has led to a considerable interest in exploiting data collected by current spaceborne satellites such as ERS-1, ERS-2 and RADARSAT for the retrieval of soil moisture information and the generation of soil moisture maps at various scales. A major difficulty in the development of soil moisture retrieval algorithms, however, has been the confounding influences of surface roughness conditions which significantly affect the relationships between radar backscatter and soil moisture. Indeed the dynamic range of σ0 due to variations in surface roughness is usually comparable to or larger than that associated with soil moisture, so that an accurate estimate of the contribution of surface roughness conditions is a prerequisite for retrieving useful soil moisture information from SAR data. This is especially true for single polarisation single frequency systems such as ERS-1 and –2 , where no independent estimate of roughness conditions is possible using satellite data alone.
In this respect, a general approach for estimating and subsequently disentangling the relative influences of soil moisture and roughness on the backscattering coefficient is to make use of physically-based scattering models. The latter yield predictions of σ0 as a function of surface conditions and can thus be used to simulate a much larger range of roughness and humidity conditions than would be encountered in experimental situations. A considerable amount of work in the past has focused on the derivation of various electromagnetic scattering models applicable to different surface conditions. For recently developed models such as the Integral Equation Method (IEM), the excellent agreement between the model results and measurements made in anechoic chambers on artificial rough surfaces [1] give some confidence in the ability of such models to accurately describe surface scattering processes. However, when applied to natural surfaces, results derived using the same models have been inconsistent so far. Thus, at present, the weak point in the development of model-based soil moisture retrieval algorithms appears to be due to an inadequate description and measurement of natural surfaces rather than a failure of the models themselves.
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For electromagnetic modelling purposes surfaces have usually been modelled as stationary randomly-rough surfaces characterised through a unique spatial scale in both the horizontal and vertical directions. A given surface is thus defined statistically using three parameters determined from the surface height profiles: the root-mean square deviation (rms) in surface height (s), the autocorrelation function and the associated correlation length (l). These three parameters (s, l and correlation function shape) represent the input parameters to the scattering models.
Profile data acquired under natural conditions has shown, however, that most bare soil surfaces exhibit large spatial variations making it difficult to determine consistent roughness parameters for use in modelling and inversion. Different surface profiles acquired in an "apparently" homogeneous field yield a large range of s and l values [2]. The most critical point in this respect is the correlation functions which are highly variable and appear to be unrelated to surface roughness conditions [2,3].
A possible explanation for these observations can be found in the measurement errors associated with profiles of finite length and finite resolution. Theoretical work, using simulated surfaces, has already been carried out in this direction. The simulation results have generally shown that the shape of the autocorrelation function is sensitive to measurement resolution [4] and that both the accuracy with which the correlation length parameter can be estimated and its precision depend strongly on the total extension of the measurements [5]. In particular, Oh and Kay [5] demonstrated that the variability in the l estimates decreases with profile length and that a mean estimate of l with a precision of ±10 % requires a profile length of 200l. Their results also show that the correlation length estimates at short profile lengths are biased towards values less than the true underlying value for l, but that this bias decreases as profile length increases and that the estimates of l approach the true correlation length. Some experimental results using profiles acquired over different European sites have shown, however, that the variability in correlation length tends to increase as profile length increases and that the observed increase in average correlation length cannot be accounted for using the classical single-scale roughness description [6].
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Because of the inadequacy of the single-scale models in describing experimental observations, a number of alternative multi-scale roughness description models have been proposed. Beaudoin et al. [7], for instance, models agricultural field roughness as a superposition of small-scale roughness components, associated with soil clods, with large scale roughness components associated with tillage patterns and drainage relief. Another approach has been to model surface roughness using random fractals [6]. In fact a number of works dealing with the mathematical characterisation of surfaces in various fields and for various applications, have shown that natural surfaces are often better described using random fractals instead of stationary single-scale processes [8,9]. For such fractal processes, rms height and correlation length parameters no longer represent intrinsic surface properties but depend on the measurement process. Contrary to the conventional single-scale roughness description model, the fractal description model predicts both a continuous linear increase in correlation length as a function of profile length, as well as a large scatter in l estimates which increases with increasing profile length.
It is within this context that, in this paper, we address the issue of roughness description models for agricultural surfaces applicable to a wide range of spatial scales. We do so by exploiting a new database of roughness profiles collected in agricultural fields at two different sites in France and Italy. The main characteristic of these profiles is that they were collected using a novel laser profiler capable of recording profiles 25 metres in length, whereas much of the profile data collected to date in the context of radar remote sensing studies has been of the order or 1 or 2 metres [2,10,11,12]. The advantages associated with the longer profiles are that: • they permit a better estimate of the characteristic roughness parameters than was possible until now; • the profiles include information on the large-scale roughness variations which allows for a better differentiation between the contrasting single-scale and multi-scale roughness description models [13]; • they provide a complete picture of the roughness properties over the spatial scale of a SAR resolution cell.
In section II of the paper, the main characteristics of the profiling instrument are described. Section III describes the test sites and the experimental data acquired. In section IV the experimental results are
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presented. We first describe the roughness classification scheme adopted. The roughness properties of the different tillage classes are then analysed in terms of rms height, correlation length statistics and correlation function shape at various spatial scales. In section V, a roughness description model is proposed which accounts for the experimental observations. The applicability of this model is demonstrated by comparing experimental and model autocorrelation functions. Finally, in section VI, the results are summarised and the implications for radar remote sensing studies discussed.
II – CESBIO-ESA laser profiler Unlike traditional laser profilers, which are characterised by high spatial resolutions of the order of 1 mm or less and relatively short profile lengths - usually less than 2m - the CESBIO-ESA laser profiler is capable of acquiring roughness profiles up to 25 metres long. The spatial resolution of the instrument is 5mm which corresponds to approximately λ/10 at C-band. The instrument is thus well suited for capturing roughness information at those spatial scales of importance in modelling backscatter from bare surfaces for current and future C-band radar satellites such as ERS-1/2, RADARSAT and ENVISAT.
In order to improve both the portability and mechanical rigidity of the profiler system, a strategy was developed of collecting carefully-aligned juxtaposed profiles, each 5 metres long, which then simulate a single long profile. This process and the different components of the system are illustrated in Fig. 1. The instrument itself consists of two 2.5 meter sections of an aluminium I-shaped beam which are joined at the middle to make 5 metres. The beam is held aloft by two special supports at each end (S1 and S2) which can be adjusted in all three directions for alignment purposes. Profile data is acquired using a laser distance meter which records the distance between the beam and soil surface every 5 mm with an vertical precision of ±1.5mm. The corresponding error in surface roughness statistics was estimated using surface simulators and found to be less than 10% for both rms height and correlation length estimates as long as the surface rms height of the surface lies above 0.5cm. This is true for almost all agricultural surfaces. The distance meter is lodged within a motor-driven chariot (L) travelling along the beam. The data is sent directly to a laptop PC which records the data on disk. Once an individual 5 meter subprofile has been acquired, the beam is
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displaced exactly 5 metres and realigned using a theodolite (T) and the bulls-eyes attached to the supports (B1, B2 and B3). This ensures that the low frequency information beyond 5 metres is preserved in the final profile. The process is repeated until 25 metres have been reached. The instrument is illustrated in Fig. 2 showing the main beam, supports, laser-holding chariot and laptop.
III - Test sites and data acquisition In order to obtain a large variation in roughness conditions and diminish the dependence of the results on a particular site, two separate sites were selected for the measurement campaigns; Marestaing in the Gers region in south-western France, near Toulouse, and Matera in the Basilicata region in south-eastern Italy. For each site a dedicated ground measurement campaign was carried out.
The initial campaign took place at the Marestaing site from May 5th to May 23rd 1998. The region is mainly agricultural in character with the main crops being wheat, corn, soybean, sorghum and rapeseed. At the time of the in-situ measurement campaign, winter crops such as wheat were already well developed, whereas crops such as corn and soybean had recently been sown or were in the process of being planted. Two large, relatively homogeneous bare soil fields about 10 hectares in size were selected for the experiment. At the beginning of the campaign both fields were relatively smooth since they had been harrowed recently by the farmer in order to break down the clods. Toward the end of the campaign, corn was sown in both fields and a large roller was passed over the field after sowing, effectively flattening any clods remaining on the soil surface.
The second campaign took place from October 6th to December 16th 1998 in Matera, a predominantly agricultural area which is devoted almost exclusively to wheat cultivation. At the beginning of the campaign, in October 1998, the fields were completely bare and had in some cases already been tilled to prepare them for the next winter wheat crop. The local agricultural practice consists of deeply ploughing fields after harvesting and then of harrowing them at least twice in order to prepare them for sowing. Three fields ranging between three and ten hectares were selected for ground truth data acquisition. Two of the fields
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were in the ploughed state during the initial measurements, the third had already been tilled once and was smoother. During the measurement period from October to the middle of December the selected fields were tilled several times and significant roughness changes observed.
In terms of roughness measurements made using the long profiler, a total of 78 profiles of 25 metres were acquired for both sites, including profiles acquired at 0 (parallel), 45 and 90 (perpendicular) degrees with respect to the tillage direction. It should be added the weather conditions during each campaign were relatively dry so that little or no variation was seen in the roughness properties of the fields apart from those imposed by agricultural management practices. In this paper only the 32 parallel profiles are considered. Ancillary data in the form of soil moisture data using Time Domain Reflectometry (TDR) and soil samples was also collected within the test fields at times coincident with ERS-1 and ERS-2 image acquisitions over the site.
IV - Experimental Results
A. Agricultural roughness categories In order to obtain an overview of the large roughness database and organise the analysis results, a consistent cross-site roughness description had to be developed. An initial step was thus to group individual profiles by their agricultural roughness state at the time of the profile acquisition. A potential difficulty in this respect is the fact that the relationship between tillage and soil roughness is influenced by other factors such as soil texture, aggregate stability and cumulative weather conditions in the period between the tillage operation and roughness measurements. However, in this case, the soil texture of the Marestaing and Matera fields was similar and belonged to the sandy clay loam or clay loam variety. Furthermore a good cross-site correspondence was observed between tillage status and rms height statistics. On this basis four distinct roughness categories could be discriminated, these being, in the order of decreasing roughness; ploughed, harrowed rough, harrowed smooth and rolled. Two harrowed categories had to be chosen because our experience during the ground truth campaign was that a farmer will usually make several passes with a
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harrowing instrument in order to progressively transform a rough ploughed surface into a smooth seedbed. Thus the “rough Harrowed” category corresponds to the first pass of the tillage instrument whereas the “smooth Harrowed” category corresponds to subsequent passes made with the same agricultural implement. Finally a “rolled” category was assigned to the two Marestaing fields after these had been sown and the soil flattened with a roller. Overall, of the 32 long profiles collected parallel to the tillage direction, 4 were collected over rolled fields, 9 over smooth harrowed fields, 8 over rough harrowed fields and 11 over ploughed fields. In some of the figures in this paper these four roughness categories are assigned the labels P, H1, H2 and R which are used to identify Ploughed, rough Harrowed, smooth Harrowed and Rolled respectively.
B. rms height and correlation length statistics As a first step in determining the roughness properties of agricultural surfaces, the rms and correlation length statistics as a function of both tillage state and profile length are examined. The overall statistics for both rms height and correlation length as a function of profile length are summarised in Table 1. The estimates were obtained by splitting the long profiles into non-overlapping segments of 0.5, 1, 5 and 10 metres in length respectively, detrending the each segment separately using linear regression and then estimating s and l for each segment. Finally the average and standard deviation was computed over all segments belonging to a given tillage class.
Looking at the s estimates first of all, we see that there is a good correspondence between tillage state and this parameter at all profile lengths. For instance, at 1 meter, the smooth harrowed and rolled surfaces exhibit rms heights around or less than 1cm whereas the rougher harrowed and ploughed surfaces have average rms values of 1.7 and 3.5 cm respectively. These values are in good agreement with those published in the literature, notably those found in [14]. The relatively low standard deviation in rms height also indicates that the estimates are consistent from one profile to the next and from one field to the next. The rms height parameter exhibits some sensitivity to profile length, especially over large differences in profile
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length. When comparing rms height values at 0.5m with those obtained at 10m there is an overall increase in s by a factor of 1.6 or 1.7.
The statistical character of the correlation length estimates on the other hand depends strongly on profile length as well as surface roughness conditions. Over short profile lengths different tillage states can be distinguished on the basis of their mean correlation value, as was the case for the rms height parameter. For instance, for 0.5 meter profiles the smoothest two tillage states can be distinguished from the rougher states since they are characterised by shorter correlation lengths of the order of 2 cm whereas the latter have average correlation lengths of 2.8cm and 3.9cm respectively. At 1 meter some distinction is still possible between tillage states but at 5 and 10 meter profiles there is a large increase in both average correlation length and scattering in the estimates for rolled and harrowed surfaces. The estimates for the ploughed surface as a function of profile length, however, are quite different in character. For instance while the estimated correlation length of rolled fields increases by a factor 40 - from 1.8 to 75.2 cm - for 0.5 and 10 metre profiles respectively, the values for ploughed fields only increase by a factor of 4, from 3.5cm to 14.3 cm. The relative scatter in the estimates at long profile lengths is also much higher for the smooth surfaces than for ploughed surfaces. It is worth noting that the observed trends for the smoother surfaces are in contradiction with the assumption of a single-scale surface since one would expect the l estimates to converge to a single representative value as profile length increases. For ploughed surfaces on the other hand, the initial increase in l and the flattening out of the estimates with increasing profile length is more consistent with a single-scale process.
A second interesting feature of the results in Table 1 is the fact that, at short profile lengths, smoother surfaces have shorter correlation lengths than rougher surfaces. In order to visualise these trends and study more carefully the relationships of s and l as a function of profile length, a plot of average correlation length versus average rms height estimated over each long profile was made and is presented in Figure 3. We observe that for 0.5 and 1 meter profiles correlation length and rms height are positively correlated. It is important to note that this trend can be detected in the data only because of the precise estimates obtained by averaging over many profile subsegments and because of the large range of roughness classes present within
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the graph. For smaller datasets, the considerable within class scatter for l values and overlap in estimates between adjacent roughness classes present in the 1 meter profiles would mask this relationship. At longer profile lengths s and l are no longer correlated and there is a large scatter in the l estimates.
Overall the observed statistical properties of s and l for short profiles of the order of 1 meter or less appear to be more consistent with a single-scale description than a fractal description. This is seen in the limited scattering of the correlation length estimates and the consistency of the estimates from one profile to the next. Furthermore because both parameters are sensitive to the tillage state of the surface and can be used individually to discriminate among different tillage states, the results indicate that the surface roughness component directly associated with the tillage state is well described using a single-scale roughness description model. A second important result for shorter profile lengths is the observed correlation between rms height and correlation length estimates. From a theoretical point of view, s and l are considered as strictly independent parameters for randomly rough single-scale surfaces and should therefore be uncorrelated unless an underlying relationship between s and l exists. A possible explanation for the observation is found when comparing clod sizes associated with the different tillage states. Smoother surfaces such as fields which have been harrowed are generally characterised by smaller clods whose extent in both the vertical and horizontal directions are less than the larger clods found on ploughed surfaces. Thus the characteristic vertical roughness dimensions and horizontal roughness dimensions expressed in terms of s and l appear related in the case of agricultural surfaces, especially when considering differences between well-defined tillage states.
The results obtained using longer profile segments on the other hand are much more consistent with a fractal roughness description. This is seen in the continuous increase of the average l estimate with profile length and the large scatter of l estimates at profile lengths of 5 or 10 metres even after averaging has been performed. It is however less evident in the rms height estimates which increase only slightly with profile length. For fractal surfaces, theoretical results have shown that, unlike the correlation function and associated correlation length, the rms height is relatively insensitive to the multi-scale nature of surfaces and can be approximated as constant over distances of several metres[6]. It is thus difficult to distinguish among
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fractal and single-scale surface roughness description models on the basis of rms height statistics. Accordingly, the rest of the paper focuses on describing and explaining the observed properties of the autocorrelation function.
C. Characteristics of the autocorrelation function In this section the characteristics of the experimental autocorrelation functions (ACF) are discussed. To this end an average correlation function was generated for each profile by splitting the profile into subsegments of 1, 5 and 10 metres and averaging the resulting ACFs. A single autocorrelation function was also calculated for each profile using the entire 25 meter profile, in order to investigate the characteristics of the ACF at spatial lags of 5 and 10 metres. The method used to compute the correlation functions follows that suggested by [15,16], and consists in taking the inverse Fourier transform of the power spectrum estimate. As a final step theoretical exponential and Gaussian correlation functions were generated on the basis of the estimated correlation lengths for each profile, and their shape compared to that of the experimental ACFs. The sampling interval for all ACFs was set to that of the profiler, i.e. 5mm.
Fig. 4 illustrates the ACF shapes obtained as a function of tillage state. The example ACFs were computed by splitting a single 25 meter profile into 1 meter segments, computing the ACF separately for each segment and averaging the resulting 25 ACFs. We observe that for the three smoother surfaces (R, H1 and H2) the experimental correlation function and the exponential model are in very good agreement, whereas the match in the case of a Gaussian ACF is not as good. For the ploughed (P) surface in Fig 4 on the other hand, the results are more ambiguous since the experimental ACF lies in between both theoretical ACFs for small spatial lags and the tail of the ACF agrees with neither model. To put such results on a more quantitative footing it was useful to assign a fit error value based on χ2 considerations, where we chose to represent the error σ in the fit by
N 2 ∑ ye − yt i i σ 2 = i =1 N −1
(1)
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with ye and yt representing the experimental and theoretical ACF estimates as a function of lag i. Here the range in lags for the calculation of σ is limited to [0,2l] in order to reduce the influence of the tails of the correlation functions. The resulting mean and standard deviation in the fit errors as a function of surface type and for different profile lengths are listed in Table 2. The results indicate that for short profiles (i.e. 1 meter profile lengths) the experimental ACFs of the three smoother roughness classes - rolled, smooth and rough harrowed – much better approximated by the exponential model and than by the Gaussian model. The low standard deviation in the approximation error σ also indicates that the fit is consistent from one profile to the next and one field to the next. For the ploughed category, on the other hand, the fit errors for the exponential model are significantly higher than for the other roughness categories and the fit is slightly better for the Gaussian case than for the exponential case.
As profile lengths increase, however, the significant increase in the error term for all surface types except ploughed ones indicates that the single-scale exponential or Gaussian correlation functions no longer agree with the observed experimental ACFs. For rolled surfaces, for instance, the mean value of σ goes from 1.3 to 12 when moving from 1 to 10 meter profiles. The standard deviation of σ also increases indicating large scatter in the goodness-of-fit. At 25 metres all surfaces except for ploughed are poorly represented by a single-scale process. There is also a slight roughness-dependent trend in the fit since the errors decrease moving from the rolled to rough Harrowed category. Looking at the shape of the autocorrelation function in Fig. 5, the large fitting errors can be traced back to the fact that the models generally overestimate the correlation at lags shorter than the correlation length. This is in contradiction with single-scale roughness models which predict a better fit between model and experimental ACFs as profile length increases. Interestingly the opposite trend - and one in agreement with the single-scale assumption - is seen for the ploughed surfaces, viz. the error in the fit with experimental ACF decreases for the exponential ACF with increasing profile length. The good fit between theoretical and experimental ACFs for ploughed surfaces at long profile lengths is illustrated in the corresponding graph of Fig 5 as well.
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V - Modelling of roughness characteristics using simulated surfaces The above results indicate that the horizontal roughness properties of agricultural surfaces, summarised through the autocorrelation function, depend strongly on the spatial scales which are considered. Furthermore it would appear that, for most of the surfaces studied, the processes affecting roughness at small scales - where the single-scale model matches observations quite well – differ from those at large scales, where conventional roughness descriptions do not apply. In this section we make use of numerically generated surfaces based on single-scale and fractal models to identify an overall descriptive model that is able to explain the observed roughness characteristics. The descriptive model is then used to generate simulated autocorrelation functions which are compared to experimental observations.
A. Fractal surface generation method A number of algorithms are described in the literature which permit the generation of synthetic fractal-like surfaces. These included the mid-point displacement technique [17] and simulations based on the Weierstrass function [18]. For the purposes of our simulations a technique based on spectral synthesis was chosen for its speed – permitting the quick generation of long profiles - and simplicity. The technique is based on the spectral representation of fractal-like surfaces which are characterised through power spectra following inverse power laws of the form [19]
S( f ) =
1 . fν
Here the one dimensional fractal dimension D is related to ν through [19]
D=
5 −ν . 2
Thus the condition to be imposed on the coefficients ak of the Fourier transform in order to obtain S(f) ∝ 1/fν is
( )∝ f1
E ak
2
ν
(2)
and the surface-generation method simply consists of randomly choosing coefficients satisfying Equation (2) and then computing the inverse Fourier transform to obtain a surface S(x) in the spatial domain.
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B. Comparison between experimental and modelled autocorrelation functions A difficulty when investigating an overall descriptive model of roughness properties at large scales (equal to those of the resolution cell for instance) is the large fluctuations of the correlation functions from one profile to the next and from one field to the next, even for fields which are similar in the agricultural or tillage sense. However when the experimental correlation functions are plotted grouped by roughness category some significant overall trends can be seen and can be explained using simulation results.
This is illustrated in Fig 6 a) which plots all available experimental autocorrelation functions calculated using 25 meter profile data and grouped by tillage state. Spatial lags up to 10 metres are considered. Since an initial analysis of the data showed that the ACFs of rolled and smooth harrowed categories had a very similar long scale behaviour, the latter have been grouped into a single “seedbed” category thus yielding three distinct roughness classes each characterised by 8 or more profiles. The average autocorrelation function for all three roughness categories, represented by the thick solid lines in each figure, is characterised by a sharp initial drop in the ACF followed by long relatively flat tails. An important feature of the average ACFs is that the strength and general shape of this drop clearly depends on the tillage state of the surface, and more specifically on its roughness. For smooth surfaces the ACF drops very little before flattening out, for fairly rough harrowed surfaces the drop is more significant whereas it represents the dominant feature of the ACF for ploughed surfaces.
A possible overall descriptive model for agricultural surface roughness which agrees with the previous observations is to consider agricultural roughness as being the result of a superposition of a single-scale process related to the tillage state of the field, and a background fractal process which is associated with natural topographic variations present in the fields. The surface can then be modelled as
S ( x, y ) = Z ( x, y ) + F ( x, y ) (3)
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where Z(x,y) and F(x,y) represent the single-scale and fractal random processes respectively, and S(x,y) the resulting surface.
In order to illustrate the overall applicability of this model, a test set of randomly rough surfaces was generated by adding a white noise process of various amplitudes to a set of fractal curves and calculating the average autocorrelation function of the curves. White noise is used in this context as a first approximation of a single-scale process. For the profile lengths considered it is a good approximation since the correlation length of Z(x,y) will generally much shorter than the total correlation length of S(x,y).
The results of this modelling process for each tillage category, along with the average experimental ACF, are given by the dash-dot lines in Fig 6 b). They were obtained by averaging the ACFs of 100 realisations of a fractal process with a step size of 0.5 cm and a total profile length of 25 metres, to which various amounts of white noise representing the single-scale component were added. The best overall agreement between model and observations was obtained by setting D, the fractal dimension, equal to 1.6 and superposing white noise with standard deviations of 40, 80 and 230% that of the fractal process for seedbed, harrowed and ploughed surfaces respectively.
Generally it can be seen that the modelled correlation function and average experimental autocorrelation functions are in good agreement. This is especially true for the sown surfaces whose single-scale characteristics appear to be well-modelled by white-noise and the “fractal” tails of the ACF well represented by D=1.6 surfaces. For harrowed and ploughed surfaces, the overall shape of the modelled autocorrelation function agrees with measurements but the initial drop in correlation for small lags is too sharp, mainly due to the white noise approximation used. A better fit would probably be obtained by using a single-scale exponential process to describe the influence of tillage roughness instead of white noise, especially given the good overall agreement of this model with observations for shorter profiles.
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VI - Discussion and Conclusions The objective of this study was to investigate the roughness properties of agricultural soils at small and large scales and to explain the experimental results in terms of an overall roughness description model. A general result was to show that the observations can be modelled using a superposition of a single-scale process related to the tillage state of the fields, and a multi-scale fractal process which is related to field topography. The relative importance of each process in determining the roughness properties of the surface depends both on the range of spatial scale (i.e. spatial bandwidth) over which the observations are collected and on the tillage state of the surface.
For short profiles around 1 meter long it is the single-scale process that dominates the roughness characteristics. At this scale the horizontal roughness properties are generally well approximated by an exponential correlation function. A second important result at this profile length was the observed correlation between s and l, which is not predicted by the single-scale roughness description theory but can be explained in terms of the relative sizes of the soil clods associated with the different tillage states. This result could be of importance in radar soil moisture inversion studies since it provides a way of reducing the number of unknowns related to surface roughness conditions from two to one, once the surface rms height has been defined. Another advantage in this respect is that a large number of rms height observations have been made whereas little data exists on correlation length [14]. However further study is needed to see how robust the relationship between s and l remains under a larger variety of conditions. At the same time it is also important to establish the range of spatial scales to which the radar measurements are sensitive for natural surfaces since these then define the scales over which the soil roughness properties need to be characterised.
For longer profiles we have seen that, for the smoother agricultural surfaces, fractal or multiscale properties of surface roughness that play a dominant role in determining the observed roughness properties of the surface. The extent to which the fractal description influences the overall shape of the autocorrelation
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function depends, however, on the relative amplitude of the single-scale process. We have seen that for very smooth surfaces such as rolled or smooth harrowed fields the overall shape of the autocorrelation function is determined mainly by the fractal process. For very rough agricultural surfaces on the other hand the shape of the ACF is mainly determined by the single-scale process related to the tillage state. In terms of radar remote sensing studies these results indicate that the incorporation of a multiscale roughness description in forward modelling and inversion is likely to be more important for smoother agricultural surfaces than for rougher surfaces.
Finally it is important to point out that the dataset analysed and the conclusions reached within the context of this study concern the observed roughness properties of only 5 fields. Moreover only agricultural fields with a well-defined tillage state have been considered. It is likely for instance that the roughness characteristics of natural bare surface differ from agricultural surfaces since they are the result of natural processes alone. While the description of agricultural roughness as a superposition of random single-scale and fractal processes appears reasonable and describes the experimental data well in a general sense, some exceptions were noted even for this limited dataset. Furthermore the roughness characteristics of agricultural fields in the direction perpendicular to the furrows have not been addressed. Considerable work thus remains to be done both in extending the characterisation of agricultural roughness to other surfaces and datasets and in assessing the impact of the superposition of single-scale and multi-scale processes in terms of the forward modelling and inversion of SAR data over bare surfaces.
Annex The CESBIO-ESA profilometer described in this paper is at the disposal of the scientific community for roughness measurements experiments. Based on a small written proposal describing the campaign to be undertaken with the profilometer, ESA will make available free of charge the profilometer to the experimenter. The cost for insurance and transportation will have to be borne by the experimenter. Please contact the authors for more information.
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References [1] A. K. Fung, Microwave scattering and emission models and their applications, Artech House, Boston, MA, 1994. [2] M. Borgeaud, E. Attema, G. Salgado-Gispert, A. Bellini, and J. Noll, “Analysis of bare soil surface roughness parameters with ERS-1 data”, in Symposium on the extraction of bio- and geophysical parameters from SAR data for land applications, Toulouse, France, 1995. [3] F. Mattia, J.-C. Souyris, T. Le Toan, D. Casarano, F. Posa, and M. Borgeaud, “On the surface roughness characterization for SAR data analysis”, in International Geoscience and Remote Sensing symposium (IGARSS’97), 1997. [4] J. A. Ogilvy and J. R. Foster, “Rough surfaces: Gaussian or exponential statistics”, Phys. Rev. D: Appl. Phys., no. 22, pp. 1243-1251, 1989. [5] Yisok Oh and Young Chul Kay, “Condition for precise measurement of soil surface roughness”, IEEE Transactions on Geoscience and Remote Sensing, vol. 36, no. 2, pp. 691-695. [6] F. Mattia and T. Le Toan, “Backscattering properties of multi-scale rough surfaces”, Journal of Electromagnetic Waves and Applications, vol. 13, pp 491-526, 1999. [7] A. Beaudoin, T. Le Toan and Q. H. J Gwyn, “SAR observations and modeling of the C-band backscatter variability due to multiscale geometry and soil moisture”, IEEE Transactions on Geoscience and Remote Sensing, vol. 28, no. 5, pp. 886-895, 1990. [8] J. A. Keller, R. Crownover and R. Chen, “Characteristics of natural sufaces related to the fractal dimension”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-9, no. 5, 1987. [9] P. A. Burrough, “Fractal dimensions of landscapes and other environmental data”, Nature, vol. 294,1981. [10] M. Zribi, O. Taconet, S. Le Hégarat-Mascle, D. Vidal-Madjar, C. Emblanch, C. Loumagne and M. Normand, “Backscattering behaviour and simulation comparison over bare soils using SIR-C/X-SAR and ERASME 1994 data over Orgeval, Remote sensing of Environment”, vol. 59, pp 256-266, 1997. [11] E. T. Engman and J. R. Wang, “Evaluating roughness models of radar backscatter”, IEEE Transactions on Geoscience and Remote Sensing, vol. GE-25, no. 6, pp 709-713, 1987. [12] L. Rakotoarivony, O. Taconet and D. Vidal-Madjar, “Radar backscattering over agricultural bare soils”, Journal of Electromagnetic Waves and Applications, vol. 10, no. 2, pp 197-209, 1996. [13] M. Davidson, T. Le Toan, Maurice Borgeaud and Terhikki Manninen, “Measuring the roughness characteristics of natural surfaces at pixel scales: moving from 1 metre to 25 metre profiles”, International Geoscience and Remote Sensing symposium (IGARSS’98), Seattle, 1998. [14] T. J. Jackson, H. McNairn, M. A. Weltz, B. Brisco and R. Brown, “First order surface roughness correction of active microwave observations for estimating soil moisture”, IEEE Transactions on Geoscience and Remote Sensing, vol. 35, No. 4, pp. 1065-1069. [15] J. Bendat and A. Piersol, Random data: analysis and measurement procedures, Wiley and Sons, New York, 1986.
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[16] W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical recipes: the art of scientific computing, 2nd edition, Cambridge University Press, 1996. [17] H.-O. Peitgen and D. Saupe (eds.) “The Science of Fractal images”, Springer-Verlag, Berlin, 1988. [18] J. T. Chen, J. M. Keller and R. M. Crownover, “On the calculation of fractal features from images, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 10, pp. 1087-1090, 1987. [19] J. A. Ogilvy, “Theory of Wave Scattering from Random Rough Surfaces”, Adam Hilger, Bristol, England, 1992
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Figures
Figure 1: Illustration of profile collection process and major components of the laser profiling system.
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Figure 2: Illustration of the CESBIO-ESA laser profiler showing the specially engineered supports, the 5 meter I-shaped beam and motor-driven chariot holding the laser distance meter, which is connected a laptop computer.
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50 cm
1m
5
8
Corr. Length (cm)
Corr. Length (cm)
4 3 2
Rolled Harrowed 1 Harrowed 2 Ploughed
1 0
6
4
2
0 0
2 4 RMS height (cm)
6
0
5m
2 4 RMS height (cm)
6
10m
40
120
30
Corr. Length (cm)
Corr. Length (cm)
100
20
10
80 60 40 20
0
0 0
2 4 RMS height (cm)
6
0
2 4 RMS height (cm)
6
Figure 3: Relationships of s and l estimates as a function of profile length. Each point corresponds to the average estimates over a single 25 meter profile.
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Figure 4: Illustration of the fit between experimental and theoretical correlation functions for profile lengths of 1 meter. The experimental autocorrelation functions in this case represent an average autocorrelation function obtained by computing the ACFs of individual 1 meter profile subsections and averaging over the full 25 meter profile length.
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Figure 5: Illustration of the fit between experimental and theoretical correlation functions for profile lengths of 25 metres.
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Figure 6: Experimental and modelled autocorrelation functions for different roughness categories. N.B. The profiles for smooth harrowed and rolled fields have been grouped into a single seedbed category. In part (a) the thin black lines correspond to individual correlation function estimates and the thick black line to the average of the experimental ACFs. For part (b) the average experimental ACF (solid line) is plotted along with the model ACF (dash-dot line) calculated using a superposition of single-scale (white noise) and fractal random surfaces.
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Tables Tillage Rolled Harrow 2 Harrow 1 Ploughed
Avg. Std. Dev. Avg. Std. Dev. Avg. Std. Dev. Avg. Std. Dev.
0.5m s 0.7 0.2 0.9 0.1 1.5 0.3 2.8 0.4
l 1.8 0.3 2.0 0.3 2.8 0.3 3.9 0.3
Profile Length 5m l s 2.7 1.0 1.3 0.2 2.9 1.2 0.8 0.2 4.4 2.2 0.7 0.3 6.1 4.5 1.0 0.9
1m s 0.8 0.2 1.0 0.3 1.7 0.3 3.5 0.6
l 10.7 6.7 12.5 9.4 14.3 7.6 11.2 1.2
10m s 1.3 0.3 1.4 0.3 2.5 0.3 4.7 0.9
l 75.2 40.0 26.3 25.2 22.9 15.0 14.3 4.2
Table 1: Average and standard deviation of rms height (s) and correlation length (l) estimates as a function of tillage class and profile length. All values are in cm.
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σ (x1000) R H2
Profile Length 1meter Gaussian Fit Exponential Fit 5 metres Gaussian Fit Exponential Fit 10 metres Gaussian Fit Exponential Fit 25 metres Gaussian Fit Exponential Fit
H1
P
Mean Stdev. Mean Stdev.
15.6 8.8 1.3 1.4
12.7 7.5 1.4 1.1
9.9 3.2 0.9 0.8
3.8 1.7 5.7 3.0
Mean Stdev. Mean Stdev.
31.6 16.5 6.5 5.9
21.4 11.3 3.8 4.4
24.8 7.3 2.7 1.6
11.4 5.1 1.5 1.0
Mean Stdev. Mean Stdev.
31.6 19.5 12.0 4.4
32.8 20.2 7.8 7.7
35.5 10.0 6.1 3.8
15.9 7.0 1.3 1.1
Mean Stdev. Mean Stdev.
23.9 14.7 34.5 32.1
39.4 27.2 19.2 18.4
37.9 16.8 15.7 10.5
24.5 15.2 2.4 2.2
Table 2: Mean and standard deviation in the fit error σ between experimental and Gaussian and exponential correlation functions.
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