2.1.

Synthetic Aperture Radar Data

For this study, we used 84 SAR single look complex images from the ERS-1 and -2 satellites, covering most parts of the catchment area. ERS-1 and -2 are twin satellites developed and operated by ESA from 1991 to 2000 and from 1995 to 2011, respectively. The SAR instrument on both satellites uses C-band ($5.6\mathrm{cm}/5.3\mathrm{GHz}$) in VV-polarization. ERS-1 and -2 operated with a fixed incidence angle of 23 deg. The SAR data were processed and georeferenced to quadratic 20-m pixels. They represent the backscattering coefficient sigma nought. The processing included speckle filtering using a $3\times 3$ Lee filter²⁸ and geometric and radiometric terrain correction using a digital elevation model with 10-m resolution.²⁹ All SAR processing tasks were performed with the software packages SARscape 5.0 (sarmap SA, Purasca, Switzerland) and ENVI 5.1 (Exelis Visual Information Solutions, Inc., Boulder).

Table 1 gives an overview of the ERS images that have been used in the analysis. Among them are images acquired in ascending and in descending node, thus incorporating different acquisition geometries. As we are only interested in relative backscatter intensities, we tested whether there are systematic differences between the distribution of means and variances of backscatter intensities for each orbit. Statistical $f$- and $t$-tests performed with R software³⁰ showed no significant differences between the distributions of backscatter intensities from ascending and descending orbits. We performed the same tests for ERS-1 and ERS-2 data. Also, these datasets showed no significant differences. Hence, we used images from ascending and descending orbits and from ERS-1 and ERS-2 jointly.

Table 1

Overview of the 84 ERS scenes. Orbital node is differentiated by asc (ascending) and desc (descending). The mean 7-day antecedent precipitation (AP7) from 10 DWD stations for each acquisition date is given in millimeters.

2.2.

Land Use Information

To account for the vegetation impact on the backscattering coefficient, Landsat 5 thematic mapper and Landsat 7 enhanced thematic mapper plus images were used to derive annual land use maps. Ground truth information on land cover, which is needed for land use classifications, has been acquired through management records supplied by co-operating farmers. These management records encompass 150 fields in total and cover different crop types. Utilizing a maximum likelihood classification, land use maps for every year were produced with a spatial resolution of 30 m. Since C-band penetrates only short vegetation types and bare soil, we concentrate on agricultural areas. Built-up and forest areas are excluded from our analysis. Within the agricultural areas, we differentiate between cereal, sugar beet, and grassland, as their roughness and dielectric properties are significantly different. Those are the main agricultural land use types in the Rur catchment, with a coverage of 15% (cereals), 13% (sugar beet), and grassland (15%) in 2001 (Fig. 1) in the area of the Rur catchment.

Maize is of minor importance compared to the aforementioned land use classes but still covers 6% of the area of the Rur catchment.³¹ Nevertheless, since the ground truth data only comprise very few maize fields, maize fields were masked from the final land use classifications. Due to missing ground truth data between 1991 and 1994 and missing cloud-free Landsat data for 1998, land use information is only available from 1995 to 1997 and 1999 to 2003. The overall accuracy of the land use maps for all years is between 82.4% (for 1999) and 99.4% (for 2001), and the kappa-coefficient is between 0.59 (for 1999) and 0.98 (for 2001).

2.3.

Soil Moisture and Precipitation Data

No in-situ surface soil moisture for validation is available for the study period and location. Therefore, precipitation data is used to evaluate the soil moisture index approach. The German Weather Service (DWD) provides daily-mean precipitation datasets for several stations within the Rur catchment over a long period. Ten stations, shown in Fig. 2, recorded data over the whole period of 1995 to 2003. They were acquired in the context of precipitation routine measurements performed by DWD and are publicly available from the Climate Data Center.³² The precipitation amounts in Fig. 2 show generally higher mean annual precipitation with increasing altitude of the station. In the eastern and western part of the Eifel region, topography-induced differences can be seen between stations.

To compare the relationship between antecedent precipitation data and in-situ near-surface soil moisture measurements in general, data from seven meteorological stations (Campbell Scientific) were used. Out of the seven stations, two provided data only in 2015 (Meteo 2 and Meteo 3) and the others provided data in 2016 (Meteo 1 and Meteo 4-7). Only data from the vegetation period (March to October) are used in the analysis. The stations recorded continuous precipitation heights and soil moisture at 5 cm depth. Volumetric soil moisture is measured with CSC616 soil moisture sensors (Campbell Scientific), using time-domain reflectometry. The stations were set up on agricultural fields from co-operating farmers in the Rur catchment. Daily precipitation sums and soil moisture means have been calculated. The measurements provided by overall eight meteo-stations were checked for consistency and homogeneity. As inconsistent data were found at one station, data from this station were excluded from further consideration. For this station, measurement errors like wrongly installed sensors are assumed.

3.1.

Soil Moisture Index

The available land use maps for each year distinguish sugar beet, cereals, and grassland pixels. All other areas are masked out, since, as explained in the data section, soil moisture retrieval with C-Band is only possible for bare soil and agricultural land use types. The land use maps for each year were resampled to 20-m resolution using a majority method and then georeferenced to a corresponding SAR scene. This was done for the ascending and descending node, respectively. To avoid mixed pixel effects, border pixels between two different land use types were masked using a dedicated filter that masks out pixels that are surrounded by more than 60% of pixels with a different land use type. Resampling and georeferencing were performed using ArcGIS (ESRI, Redlands).

For each land use group (cereals, sugar beet, and grassland) and month, histograms of the sigma nought values in dB were produced. The soil moisture index was then scaled to the 5th and 95th percentiles of each histogram. Pixels outside of this range were discarded. Limiting the histogram analysis to these percentiles reduces spurious effects due to speckle or outliers. The interval between the two percentiles is then used to normalize the sigma nought values based on their position in the histogram. The soil moisture index (SMI) value for a given land use and month combination is calculated similar to Wagner et al.²⁴ as follows:

3.2.

Evaluating Soil Moisture Index Maps

Typically, ground measurements taken during the satellite overflight are used to validate remotely sensed surface soil moisture data. However, for historic data like ours, these measurements are usually not available. Therefore, we used the following two step approach to validate the SMI: (1) by comparing soil moisture index maps of two consecutive days from ERS tandem acquisitions and investigating the drying and wetting behavior using precipitation measurements and (2) by correlating the qualitative soil moisture index maps with antecedent precipitation that was measured at DWD stations.

From August 17, 1995 until June 02, 1996, ERS-1 and 2 operated in the so-called tandem phase. During this phase but also beyond, several acquisitions were made from the same path with 24 h delay between the two satellite overpasses. Those tandem scenes can be used to observe short-term changes in soil moisture like drying or wetting of the soil surface. Other parameters that may influence the backscatter intensity (surface roughness and vegetation cover) can be assumed constant on such a short-time scale. Seven of those tandem scenes are available for our area and observation period. By calculating the differential SMI ($\mathrm{\Delta }\mathrm{SMI}$) from two consecutive days and comparing it to precipitation data, we evaluated the resulting SMI maps in terms of soil moisture change. As the SMI comprises the values between 0 and 100, $\mathrm{\Delta }\mathrm{SMI}$ can take integer values between $-100$ (drying) and 100 (wetting).

Another way to validate the soil moisture index approach is to use antecedent precipitation data as a comparable quantity. Antecedent precipitation is used as an approximation to soil moisture when soil moisture data is unavailable.^37,38 As antecedent precipitation is based on measurements at meteorological stations, its spatial resolution is rather coarse. Substituting or complementing antecedent precipitation with SMI would result in an improved spatial representation. However, this requires that a strong relationship between the soil moisture index and antecedent precipitation can be found.

For this analysis, 10 DWD precipitation (RR) stations (Fig. 2), which provide a complete time series of measurements and are located within our test area, were chosen. The mean SMI value for a region encompassing a 1-km radius around the DWD RR stations was calculated.

3.3.

Geostatistical Analysis

One major purpose of our study is to understand the spatial heterogeneity of surface soil moisture. Spatial heterogeneity can be expressed by measures of spatial variability and spatial structure (e.g., the autocorrelation structure).^39,40

Patterns of spatial variability were examined by analyzing the distribution of standard deviation throughout the Rur catchment. Therefore, standard deviation and mean SMI values were calculated from the single pixel values in $140\mathrm{m}\times 140\mathrm{m}$ nonoverlapping boxes. Only boxes that contain less than 30% of masked-out pixels were considered in the analysis.

The spatial structure was assessed using semivariograms. The experimental semivariogram describes how data correlate at distinct distances (given by the lag). The shape of a semivariogram can be described by three parameters: range, sill, and nugget. To parameterize their values, a curve, the theoretical semivariogram, is fitted to the experimental semivariogram. The range parameter can be used to describe the break between spatially correlated and uncorrelated values and hereby provides information about the spatial organization of soil moisture patterns.⁴¹ The sill describes the variance of uncorrelated values and the nugget describes small-scale variability and measurement error. Patterns in those variables can reveal patterns in the distribution of surface soil moisture. Experimental semivariograms have been calculated using the R package gstat,⁴² utilizing a maximum lag of 25 km and a lag class size of 200 m. To fit the theoretical semivariograms, the ordinary-least-squares method has been used, employing an exponential model.

4.1.

ERS-1 and -2 Tandem Scenes for the Evaluation of Drying and Wetting Processes

For the seven paired scenes of the tandem acquisition (Table 3), $\mathrm{\Delta }\mathrm{SMI}$ was calculated. On the map from March 26, 1996/March 27, 1996 (Fig. 5), we clearly see strong drying (red), whereas for April 30, 1996/May 01, 1996, we see wetting (blue) of the area. This is well in line with the measured daily precipitation at the 10 meteorological stations (Table 3). For the drying case, precipitation has been recorded at all stations on March 26, 1996. One day later, on March 27, 1996, only two stations show precipitation. This correlates with the “drying” indicated in the differential map. For the wetting case, no precipitation has been recorded at the stations on April 30, 1996, whereas on May 01, 1996, all stations recorded precipitation. Again, $\mathrm{\Delta }\mathrm{SMI}$ captures this development well. Looking at all seven tandem pairs, statistical $F$-tests and Levene-tests between each tandem-pair showed inequality in variance for all cases. The subsequent Welch-tests showed significant differences in mean for each tandem pair. This indicates a clear change in the backscatter distribution from one to the next image in the tandem pairs. The mean $\mathrm{\Delta }\mathrm{SMI}$ and the number of pixels in each differential image, which display drying or wetting, were compared to precipitation at the different stations (Table 3). A positive tandem mean $\mathrm{\Delta }\mathrm{SMI}$ shows that the differential image illustrates a wetting tendency and a negative tandem mean $\mathrm{\Delta }\mathrm{SMI}$ shows a drying tendency. The effect of precipitation is well represented by the change in the tandem images, which can be accredited to a change in soil moisture. Precipitation situations without a clear trend, as on October 03, 1995/October 04, 1995 and on May 20, 1997/May 21, 1997, result in an almost equal percentage of pixels showing wetting and drying tendencies and the mean of the differential tandem image is close to zero. In general, on dates where a drying or wetting can be expected from precipitation, this tendency can also be found in the $\mathrm{\Delta }\mathrm{SMI}$.

Table 3

Mean and standard deviation of precipitation (mm) as measured at different stations for dates of tandem acquisitions with corresponding mean SMI difference (ΔSMI) and the percentage of wetting and drying pixels.

Fig. 5

Differential images ($\mathrm{\Delta }\mathrm{SMI}$) for two ERS tandem pairs with corresponding precipitation for both days, of each pair, respectively, at the 10 DWD stations introduced in Fig. 2. Note that April 30, 1996, was a dry day everywhere in the test site and considerable rainfall fell on May 01, 1996. The opposite is true for the tandem scene March 26, 1996/March 27, 1996.

4.2.

Comparing the Soil Moisture Index to Antecedent Precipitation

The previous section showed that the SMI represents wetting and drying tendencies due to precipitation well. In this section, we investigate whether the SMI shows a similar behavior with regards to antecedent precipitation as an in-situ soil moisture dataset does. So, SMI mean values at each station and in-situ soil moisture data were compared to corresponding antecedent precipitation sums.

The general relationship between antecedent precipitation and surface soil moisture is influenced by different factors as meteorological conditions (e.g., temperature and humidity) that regulate the evapotranspiration and also by interception and saturation state of the soil that are not considered in our analysis. Depending on these factors, the same antecedent precipitation can cause different soil moisture and thus different SMI values. Therefore, from a theoretical point of view, a strong correlation between SMI and antecedent precipitation is not to be expected. The relation of antecedent precipitation to soil moisture has the shape of a triangle (Fig. 6). Depending on the saturation state of the soil, low antecedent precipitation can result in low to high soil moisture. High antecedent precipitation can only result in high soil moisture unless there is sealing or crust formation of the soil.

Fig. 6

Theoretical relationship between soil moisture and antecedent precipitation. The gray triangle shows the theoretically plausible area of values.

To check if in-situ measured soil moisture values (Fig. 2) fulfil our theoretical scheme, we compared daily measurements from seven soil moisture sensors at meteo-stations at 5-cm depth with 7-day antecedent precipitation recorded at the same stations (Fig. 7). The measurements are from periods in 2014 to 2016. Only measurements from the general vegetation period are used. The scatterplot fulfils the theoretical expectations and resembles Fig. 6.

Fig. 7

Comparison of 7-day sum of antecedent precipitation with daily mean soil moisture from seven meteo stations within the Rur catchment. Continuous measurements were taken in 5-cm depth. Data are from vegetation periods in 2014 to 2016.

A comparison of soil moisture index and 7-day antecedent precipitation at the 10 DWD stations and the soil moisture index in a 1-km buffer around these stations (Fig. 8) also shows a behavior consistent with theoretical considerations and resembles Fig. 7. Low amounts of antecedent precipitation cause low to high SMI values but high amounts of antecedent precipitation do not cause low SMI values. The fact that both SMI and in-situ measured soil moisture react to antecedent precipitation in a similar way, which is additional evidence that the SMI in fact shows near-surface soil moisture state.

Fig. 8

Comparison of 7-day sum of antecedent precipitation with mean SMI for all DWD stations and ERS scenes.

As mentioned before, no clear relationship between antecedent precipitation and near-surface soil moisture can be expected because of the many interacting factors that are involved (meteorological condition, saturation of the soil, etc.). To reduce this natural “scatter,” we averaged the antecedent precipitation values according to SMI classes using 100 and 10 classes, respectively. Thus, the SMI class of 50 represents SMI value between 49.5 and 50.4 and in the case of 10 classes, SMI class of 5 represents SMI values between 44.5 and 54.4. For each SMI class, the antecedent precipitation was averaged. Figure 9 shows the resulting relationship of average antecedent precipitation height and SMI class. The resulting linear correlation is highly significant and shows an $R^2$ of 0.59 for 100 classes and 0.92 for 10 classes. Previous research found that five to ten classes of soil moisture are distinguishable with ERS data, as ERS-based soil moisture estimates have been found to provide an RMSE of about 5 to 10 vol.%.^19,50

Fig. 9

Comparison of 7-day sum of antecedent precipitation averaged by SMI class with mean SMI for all stations and ERS-scenes. With (a) 100 classes ranging from 0 to 100 and (b) 10 classes ranging from 0 to 10.

Extreme SMI values between 0 and 10 and between 90 and 100 are very infrequent in the dataset. This did not change significantly when spanning the SMI between the 10th and 90th percentile instead of the 5th and 95th percentile (see Sec. 3.1). Very low and very high soil moisture is rarely found, especially when averaging over a larger area as we did within the 1 km buffer around the meteo-stations. This does not mean that our assumption of finding wet and dry pixels is incorrect, but it means that when averaging over a larger area, very dry and very wet situations are under-represented.

No improvement has been seen when calculating the correlation between soil moisture index and antecedent precipitation restricted to the crop areas and restricted to grassland areas, respectively. From that, we can conclude that there is no clear evidence that the soil moisture index performs different on one of the two land use types despite their different biomass development.

We performed the same analysis using an antecedent precipitation index based on Mattia et al.⁵¹ instead of antecedent precipitation and found comparable results.

The two evaluations presented in this section show that the SMI is able to display changes in the moisture state and behaves like in-situ soil moisture when compared to antecedent precipitation. Of course, low antecedent precipitation can still result in a high SMI, when, for example, evaporation is low and the soil is saturated. But, on average, low/high antecedent precipitation results in a low/high SMI, respectively, and a higher antecedent precipitation provokes the assignment of a higher soil moisture index value. Therefore, general patterns of SMI should be reliable although single pixel values are uncertain.

4.3.

Analysis of Spatial Heterogeneity

Having a time series of SMI values at high spatial resolution provides the opportunity to examine the temporal or seasonal evolution of the spatial soil moisture structure and to find possible dependencies on soil moisture state. To assess which factors regulate the spatial distribution of surface soil moisture in the Rur catchment, we investigated the spatial heterogeneity of soil moisture and its structure.

It has been found that the theoretical relationship between standard deviation and mean soil moisture shows a unimodal shape, where standard deviation is highest at soil moisture values in the mid-range between field capacity and wilting point and is low in very wet or dry areas.^1,20 The reduction in standard deviation of soil moisture under wet and dry conditions is due to the physical limits of saturation (100%) and complete dryness (0%). Thus, the spatial variability of soil moisture at extreme dry and wet states is controlled by spatial variability of soil properties, such as wilting point and porosity. When the soil reaches intermediate moisture conditions, spatial variability of soil moisture is regulated by spatial differences in the fluxes, especially of evapotranspiration. As evapotranspiration is highly heterogeneous in space, this provokes an increase in variability of soil moisture. Likewise, spatial differences in infiltration will have a regulating effect on the spatial variability of soil moisture. Spatial differences in infiltration depend to a high degree on spatial differences of soil texture. Since the soil texture in our research catchment is mainly homogenous (apart from the northernmost part of the catchment), we assume the influence of spatial differences in infiltration on spatial soil moisture variability secondary. When the soil dries further, the moisture distribution is influenced by spatial differences in the wilting point (which is less heterogeneous than evapotranspiration), thus decreasing the variability.^1,20

To investigate whether similar behavior can be found in the SMI data, mean SMI value and standard deviation have been calculated for the whole Rur catchment. We chose $140\times 140\mathrm{m}$ nonoverlapping boxes for the calculations since, in this small area, antecedent precipitation heights, soil types, and land use types are assumed to be homogenous. At intermediate mean SMI values, standard deviation shows a wide range of values up to a maximum standard deviation (Fig. 10). In dry areas (with low SMI mean) and wet areas (with high SMI mean), the range of standard deviations is low and no high standard deviation values are found. Thus, the outline of the distribution follows the findings mentioned above and shows the theoretically expected behavior of a soil moisture dataset.

Fig. 10

Comparison of SMI standard deviation and SMI mean over the whole Rur-catchment and over all dates. The analysis has been performed in $140\mathrm{m}\times 140\mathrm{m}$ nonoverlapping boxes. The line shows the theoretical relationship between standard deviation and mean of soil moisture, as described in Bell et al.¹ and Vereecken et al.²⁰

This behavior is important, as it contradicts the argument that the soil moisture index could show effects of vegetation instead of soil moisture. If we assume for a moment that the soil moisture index reacts mainly to a vegetation signal and not a soil moisture signal, the increasing variability of the vegetation’s state that emerges during the vegetation period would increase the variability in roughness and/or water content. This should lead to a increased backscatter variability (and hereby to an increased SMI variability). The relationship should probably saturate at a certain density but it would not decrease again. The decrease of variability at higher values of SMI in our data indicates that the soil moisture index method, despite its crude way of minimizing the vegetation influence, represents soil moisture and not vegetation water content or interception.

Semivariogram analysis reveals spatial patterns in the data and enables one to find causes of these patterns. In previous studies, a semivariogram analysis of surface soil moisture revealed temporal courses and dependencies on moisture state for range and sill values.^22,41 For the current study, experimental and theoretical semivariograms were generated for the three different spatial units: (i) the whole catchment, (ii) the low mountain range of the Eifel, and (iii) the fertile loess plain. Most semivariograms could not be fitted well by a conventional model, so that a nested approach had to be applied (Fig. 11), where two semivariograms are added, resulting in a short and a long range. This implies that there are at least two spatial patterns that have an influence on the SMI in the Rur catchment. Theoretical semivariograms were fitted for lags up to 10 km. In those cases, where simple and nested approaches failed to fit the experimental semivariogram, an exponential fit was used but restricted to a maximum lag of 3 km as only this resulted in acceptable fits. For the whole catchment, this was the case for 4 out of 84 dates, for the Eifel region, the simple exponential fit had been applied to 3 dates but on 4 dates, no model could be fitted at all. For the fertile loess plain again, all dates could be fitted, but for 16 out of 84 dates, the simple exponential fit has been chosen. This means that for these dates, we find several underlying influences at distances of more than 3 km that cannot be described. No commonality regarding season or soil moisture state could be found among these dates.

Fig. 11

Exemplary empirical semivariogram with simple exponential (orange) and nested exponential fit (green) showing June 06, 1997. The maximum lag for fitting has been set to 10 km. The gamma value or semivariance describes the variance in the spatial relationship and is inverse to autocorrelation.

The short ranges for the whole catchment span between 264 and 606 m, for the fertile loess plain between 276 and 672 m, and those for the Eifel region between 246 and 783 m. These scales correspond to typical field sizes found in the Rur catchment, so that we assume changes in agricultural management from one to the next field (e.g., crop type, phenological state, cutting dates on grassland) and thereby changes in evapotranspiration, are the dominating influence on the spatial structure of soil moisture on the smaller scale. Comparable ranges have been found by Reichenau et al.⁴⁰ when analyzing the spatial structure of leaf area index (LAI) on arable land in the fertile loess plain.

To find factors that influence the spatial structure of soil moisture, we correlated the ranges determined in the semivariogram analysis with mean soil moisture and its standard deviation. We also searched for temporal patterns, e.g., distinctive features in specific years or months or specific temporal courses. Short ranges are compared to standard deviation of soil moisture (which correlates with sill) in Fig. 12. For the whole Rur catchment and for the fertile loess plain, a slight negative trend, though highly variable, can be observed. This trend can be described by a negative linear correlation with an $R^2$ of 0.14 for the whole Rur catchment and an $R^2$ of 0.20 for the fertile loess plain. Both relationships are highly significant with a $p$-value $<0.001$. The negative trend is reasonable as we expect to find a faster decrease in autocorrelation with distance, the more variable soil moisture is. For the Eifel region alone, no significant trend is found in the data. The short range values for the Eifel region are, in general, slightly higher but also more variable than in the other two areas. This is explained well by the fact that the Eifel is mainly characterized by grassland areas that usually have no clear field borders but span larger areas and are influenced by heterogeneous terrain. Statistical $f$- and $t$-/Welch-tests have shown that the variance is equal for the range values calculated from the fertile loess plain and the whole catchment and also for the values from the Eifel region and the whole catchments. The mean values for both samples are unequal. The range values from the fertile loess plain, though, show inequality in variance and mean when compared to the range values from the Eifel region, indicating that the short range values calculated for the fertile loess plain and the Eifel region are substantially different. A comparison of short ranges with mean SMI neither shows a trend neither for the whole Rur catchment nor for the fertile loess plain or the Eifel region separately (Fig. 13). Thus, in our data, the spatial structure of SMI is not correlated with the mean soil moisture state. This leads to the assumption that the SMI is too variable to be represented by a single value for the whole catchment (or fertile loess plain or Eifel region). Therefore, a comparison between such mean value (that should represent the general moisture state of the observed area) and the short ranges might not be meaningful. The temporal evolution of the short range values is analyzed in Fig. 14. Here, no trend can be recognized either. This could be accredited to the low temporal frequency of the data, which makes small-scale changes invisible. What can be seen is that, in 2003, the short ranges do not reach values as low as in the other years. This might be an effect of the 2003 European heat wave that also struck Germany. Due to the drought, the vegetation influence on the spatial structure might have decreased, resulting in missing low short range values.

Fig. 12

Relationship of short range to standard deviation of SMI for the fertile loess plain, Eifel, and the whole catchment.

Fig. 13

Relationship of short range to mean SMI for the fertile loess plain, Eifel, and the whole catchment.

Fig. 14

Temporal evolution of short ranges in the whole Rur catchment, the Eifel region, and the fertile loess plain.

Korres et al.⁴¹ examined remotely sensed soil moisture products of the fertile loess plain of the Rur catchment with a spatial resolution of 15 and 150 m. They found mean ranges at 432 m at 15-m resolution and 711 m at 150-m resolution using an un-nested approach. These range values were associated with land use structure. In contrast to other datasets used in the study, no dependence on soil parameters could be identified with the semivariogram analysis for the remotely sensed soil moisture. The results of Korres et al.⁴¹ are in line with the results of the current study.

Analyzing the long ranges for the cases, where a nested fit has been applied, is more complicated, as these ranges vary a lot, up to maximum long ranges found at about 1500 km, when the theoretical semivariogram does not converge optimally. In median, anyways, the long range is found at a length of about 12 km. The 10th and the 90th percentiles of the long range distribution are 5.4 and 117.6 km, respectively, thus spanning a realistic range. Influences could be attributed to soil type, topography, or local precipitation but cannot be clearly distinguished. The long ranges found in this study are higher than those found by Reichenau et al.⁴⁰ but at a comparable scale. They also attributed these ranges to effects of soil or weather.

In summary, the only clearly recognizable effect on the spatial structure of SMI in our study is the field size. This has been found as a major influence in the Rur catchment before Reichenau et al.⁴⁰ and Korres et al.⁴¹ As we had the opportunity to analyze a long time series of soil moisture data, we had expected to find temporal or seasonal trends in the soil moisture structure due to weather or climate influence on soil moisture distribution or a dependency on the soil moisture state like for soil moisture variability, since such effects have been found in soil moisture datasets before Western et al.²² and Korres et al.⁴¹ Contrary to expectation, no such trend or dependency has been found in our data. Since Korres et al.⁴¹ found a dependency of range on soil moisture variability for the same research catchment, we had expected to find something similar. Also, we expected to find a clear signal of the two different landscape units, the fertile loess plain and the Eifel. Our results indicate that the soil moisture index, at least with this spatiotemporal resolution, is not suitable to identify more complex factors that influence the structure of soil moisture in our catchment. But also the fact that we do not have spatially continuous data, since land use types like, e.g., forests, were masked out, might not allow us to find more structure in the data.