The study area is the Attert catchment, located in midwestern Luxembourg and partially in Belgium (see Fig. 1). It is the main test site of the German DFG research project CAOS (“Catchments as Organised Systems”; CAOS, 2014) with a total catchment area of 288 km2 at the gauge in Bissen. The undulating landscape with a mean slope of 8.4 % spans between 222 m and 535 m a.s.l. The northern slopes are geologically defined by schists from the Ardennes massif, while the mainly southern slopes arise on sandstones from the Paris basin Mesozoic deposits (compare Fig. 9). Soils vary between sand and silty clay loam. The land cover of the catchment is predominantly cultivated; 4.8 % of the area is accounted for settlements and rather impermeable surface, 65.4 % for agricultural used land, located predominantly on the knolls, and 29.7 % for forests, located predominantly in the V-shaped valleys (compare Fig. 9). Climate is characterized by mean monthly temperatures between 18 ∘C in July and 0 ∘C in January (1971–2000). The mean annual precipitation is 850 mm and the mean annual actual evapotranspiration is 570 mm (1971–2000), resulting in a pluvial oceanic regime with low flows within July to September due to high summer evapotranspiration, and high flows mainly from December to February.

The multispectral imaging system ASTER on board the TERRA satellite, launched in December 1999, orbits on a near circular, sun-synchronous path with a repeat cycle of 4–16 days. The ASTER instrument consists of three sensors (VNIR – visible-near infrared: 0.52–0.86 µm; SWIR – shortwave infrared: 1.6–2.43 µm; TIR – thermal infrared: 8.125–11.65 µm) with four, six and five bands, respectively (Fujisada, 1995). For this study, only the Level 1A (raw) TIR data band 13, within 10.25–10.95 µm, with a spatial resolution of 90 m, are used. This band is chosen due to the lowest absorption of the atmosphere and therefore least altered thermal signals (compare Elder and Strong, 1953). The local overpass time is around 11.40 a.m. CET. Between January 2001 and June 2012, a total of 28 snow-free images (see Fig. 2, after preprocessing), with a maximum cloud cover of 15 %, were extracted. In addition, Corine land cover (EEA, 1995) updated from 2006 (Fig. 9, upper right), and a geological map based on dominant rock formations (SGL, 2003) (Fig. 9, lower right), are used for further analysis.

The used Level 1A (raw) TIR data product lacks a proper georeferencing. This was applied manually with 60 to 70 ground control points (depending on the cloud cover), achieving a mean accuracy of 40 m within the Attert catchment. In this transformation step, the spatial resolution of the images was adjusted from 90 m to 15 m by assigning the nearest neighbor values. The geo-positioned images were then converted from unprocessed digital numbers to top-of-atmosphere (TOA) temperatures (TTOA) with standard parameters, as given by CESSLU (2009). Sensor decay was not taken into account as decay errors due to spatially homogeneous and heterogeneous degradation of the sensor (sensitivity) are a magnitude smaller than measurement accuracy, according to Hook et al. (2007). Merely homogenous atmospheric conditions throughout the catchment were assumed for each single time step and, as our focus is on statistical pattern analysis rather than absolute LST values, atmospheric correction was omitted here and TTOA is used in the following. Additionally, calculating cloud masks was omitted as heavy fragmentation of the full time series would occur if masks were applied for even small clouds in every affected image and cumulatively applied for the full series. In the further statistical analysis, the distortion of results due to clouds is negligibly small, as occurring clouds are neither repeating in certain areas nor of large spatial extent per image. The time series of LST for individual pixels in the data set hence include one outlier due to clouds at most. This does not heavily influence further calculations on the full pattern. For simplification reasons, the calculated data are further referred to as LST time series.

The first aim was to demonstrate that LST patterns, although changing throughout time, persist to a certain degree and hence provide information on the local organization of land surface energy and water balance within the full catchment. The absence of persistency would imply competing patterns within the time series and hence sever changes within the controlling features or even oscillating states within the time series. A further investigation of the timing of the pattern changes and appropriate splitting of the time series would be imminent to a comprehensive pattern analysis. In such a case, the following steps need to be executed for the separated data sets. In order to analyze the overall pattern persistency within the time series while retaining spatial patterns, a procedure similar to that used for “co-referencing” different ASTER TIR bands is used (Hirschmüller et al., 2002). The correlation of shifted windows within two images indicates whether there is a clear shift within the overall pattern in any spatial direction, or if “blurring” occurs and persistency is then absent. Therefore, the square window w of defined size (e.g., 3 × 3 pixel (px)) around the Pc pixel of the image I1 (time step 1) is selected and the correlation coefficient is calculated for the same window (e.g., from 32=9 values) in the image I2 at time step 2 (Fig. 3a). The window within the second image is now shifted within defined maximum ranges r1 and r2 (e.g., r1=[-3,+3] in N–S direction, r2=[-3,+3] in E–W direction; Fig. 3b), and correlation coefficients are assigned for any shifted position (dx,dy) of Pc and they produce square fields of correlation coefficients (e.g., 7 × 7 px; Fig. 3c).

The persistency of the patterns in the LST data within two time steps is then assessed by calculating average correlation coefficient fields for a sample of well distributed central pixels, depending on the ratio of window and shift size to image size (to reduce the effort of calculating a shift for the whole image). The overall persistency of the patterns is the average of the correlation coefficients for all combinations of patterns within the time series (28×(28-1)=756). In case the maximum correlation coefficient is within a shift of (0,0) and the decrease of the correlation coefficients is large towards bigger shifts (= no blurring of a single peak), the persistency of the overall pattern over time is considered as high.

For our LST time series, the observed overall patterns are stationary persistent in general. By calculating the mean correlation coefficient within the full time series data set and a range of shifts of [-50,+50] in both directions (Fig. 4), it is shown that the peak correlation value is within a shift of (4,1) px and hence within the range of the resolution of one original ASTER pixel (4 × 15 m = 60 m). Also, the overall positioning of temperature values within the patterns is correlated over times, and, as a first result, it can be derived that temporal trends within the thermal images of the Attert catchment can be considered as “spatially stationary persistent”.

In addition to the overall persistency, the temporal dynamics of local LST patterns are investigated using a second type of “moving window” approach. To analyze the spatial relationship of each pixel within its local neighborhood, for each pixel Pc within an image a square window w (the environment) of a defined size (e.g., 3 × 3 px) around this central Pc is compared to the value of Pc. The environment information (ENV) is summarized to statistical information in the form of percentages of values within the square window that are bigger than, smaller than or equal to the value of Pc (see Fig. 5a for an example analysis of values that are bigger than Pc).

The variations of the ENV information over time were analyzed for the 28 LST images via the spatial assessment of the coefficient of variation (|σ/μ|) for each of the three setups (<,=,>; see example in Fig. 5c–d). The three spatially distributed coefficients of variation are finally reduced to an average pattern of coefficients of variation by taking the mean value of the three setups (Fig. 5b, right).

Low coefficients of variation over time indicate a very “stable positioning” or rank of that particular pixel within its local environment. An extreme value of zero would mean no change of dynamics over time for the pixel environments; for a value of 1, the standard deviation is as large as the mean value, suggesting that the persistency of the local pattern is rather low, and values larger than 1 have to be interpreted as non-persistent. In this way, areas of low coefficients indicate stable, persistent local patterns, and distinct varying behavior can be well identified by areas of high coefficients of variation.

The analysis of the LST time series using a window size of 15 × 15 px = 225 × 225 m2 identifies relatively low coefficients of variation (Fig. 6) with 90 % of the values between 0.19 and 0.55, 50 % within the range of 0.27 and 0.42, and only 0.03 % of the values larger than 1. This indicates a high local pattern persistency.

Based on both, global and local persistency analysis, relatively stationary patterns at the catchment scale, accompanied by stationary dynamics at the scale of hill slopes throughout the catchment can be expected. The existence of LST pattern persistence also suggests some structured control on LST by some land surface characteristics. In the following section possible controls will be extracted and analyzed.

Applying principle component analysis (PCA; for a full mathematical description, see Richards and Jia (2006; chapter 6.1)), or empirical orthogonal functions (EOFs; e.g., Denbo and Allen, 1984; Hamlington et al., 2011; Lorenz, 1956) allows the assessment of independent structures within complex data sets. Because both approaches share a similar methodology, here, PCA is used to determine which spatial factors are controlling patterns of LST within the time series. PCA uses orthogonal transformation to calculate a composition of linearly uncorrelated values of decreasing dominance from possibly correlated monitored variables. In remote sensing, PCA is often applied to reduce the number of (correlated) variables within classification procedures (see, e.g., Crósta et al., 2003; Moore et al., 2008, for the analysis of multi-spectral, single temporal TIR data to assess different geological structures).

Here, the aim is to transform the observed 28 LST patterns into patterns of virtual and independent principal components. These components represent the most dominant controlling factors for the temporal dynamics of LST pattern in decreasing order. An illustrative example for a PCA application in this context is given in Fig. 7 for artificial data.

The PCA application for the ASTER TIR time series produced 28 independent components as summarized in Table 1. By construction, components with higher (lower) degree show less (more) information and more (less) noise. 61.9 % of the variation is cumulatively expressed via the first five components (third row), while still more than 3 % of the variance are expressed by particular components (second row). In the following, a focus is given to the first five components (Fig. 9).

Figure 8 illustrates a distinct degree of structured heterogeneity for these five components. In principle the patterns of the PCs would allow to classify the catchment or landscape into different functional units that, when using LST images, would strongly reflect the functioning of the landscape related to the water and energy balance under radiation driven conditions. The number of PCs to be considered in such a classification would depend on the overall number of units that should be differentiated (which will strongly depend on computational resources available to explicitly represent within catchment variability), but also on the (cumulative) percentage of explained variance of the PCs, as well as on the distribution or, at least, range of the component values of each individual PC.

However, while this is an important topic related to land surface hydrological modeling, the focus here will be on the relationship of the extracted PCs with other land surface characteristics. Given the controls of LST as discussed in the introduction, it is expected to find some relationship of the first dominant PCs with vegetation, soil, geology, elevation, slope, aspect or others. A comparison of the PCs with available data suggested a strong relationship between PC1 and land use data, as well as PC2 with geological information. These relationships are illustrated in Fig. 9, where maps of PC1 and Corine land cover as well as PC2 and a geological map of the Attert catchment are shown next to each other.

A more detailed analysis is given by Fig. 10, where the distributions of component values of PC1 for the individual Corine land use data (Fig. 10a) and of PC2 for the individual geological classes (Fig. 10b) are plotted separately. The diagrams underpin a strong relationship between both components and suggested land surface characteristics. Concerning land cover, low component values of PC1 are shown for artificial areas, medium values for agricultural areas (arable, pastures, complex cultivation and agricultural/natural) and high values for forests. In this way, PC1 might be interpreted as related to similar dynamics in leaf area index (LAI; see Asner et al., 2003), and therefore the potential for water vapor and energy exchange between the land surface and the atmosphere. The high values for “mineral extraction” can be explained, as the single, relatively small area is surrounded by forests and partially replanted with smaller trees or shrubs during the observed time span.

When analyzing the component values of PC2 for the different geological classes, schist areas show distinct, different distributions compared to the other (mainly) sandstone areas. Schists with a high proportion of fractures are known for a high water drainage potential compared to the remaining sedimentary geology classes (see Chiang, 1971). The availability of water for transpiration and therefore the splitting of available energy into sensible and latent heat fluxes, resulting in different land surface temperatures, are thereby strongly affected. In this sense, PC2 can be interpreted as being related to bedrock information or coupled soil texture.

Even though land surface temperature is expected to depend on elevation and other terrain properties, no correlation for PC3 to PC5 (and higher) could be found with any other available observable land surface characteristic pattern and, in particular, to DEM related variables. For the Attert catchment, the elevation differences are moderate, and higher altitudes are related to the Schist areas (see Fig. 1). Thus, some part of a possible elevation effect might be “hidden” in PC2 already. However, for other more mountainous areas, possible relationships might be more pronounced and should be considered and analyzed in detail.

In addition to the component values, PCA also provides information on the weight of each component within each single time step through calculation of the specific loadings. Table 2 illustrates the first five components and their loadings for the analyzed data set. While some dependencies of the sign, mean and standard deviation of the loadings with meteorological or hydrological conditions or states in the Attert catchment are expected, here, only the differences in the loadings at individual dates are used to identify a limited number of images that are most distinct in their information content but that represent the wide range of LST dynamics over the considered time period. Based on the cumulative Euclidean distance of loadings within the LST time series, a number of 5 exemplary images are selected for further analysis (15 February 2003, 17 May 2004, 24 May 2004, 27 May 2005 and 27 March 2012).

In the following, the temporal dynamics of LST data are analyzed in terms of their “functional behavior” and, to classify the catchment into areas that show some similarity in this behavior (functional units). Similar to the analysis of pattern dynamics persistency, the vast data variability is transformed into simple information. Using the five most different images and therefore time steps (see Sect. 3.3), the data are binarized using an approach suggested by Hauhs and Lange (2008). The pixels of each image within the time step are separated into values larger than the median value of the image (1) or lower (0) (Fig. 11, left). The set of five binarized images can be aggregated into five-letter “words” (Hauhs and Lange, 2008) by concatenating these binary values (see the three-letter example in Fig. 11; right). The order of letters within the words represents the response of the land surface to differences in the water and energy balance for each pixel. These different land surface responses refer to differently behaving landscape units.

The transformation of the five LST images into behavioral words results in a (still manageable) number of 32 (= 25) classes throughout the catchment, as illustrated in Fig. 12. In some areas, functional behavior changes over short distances, indicating different response of the land surface towards radiation-driven conditions; other areas behave very similarly over larger spatial extent. These larger clusters are characterized by a constant behavior throughout the subset time series with short interruptions only (e.g., class “00010” only has one short “break” of length 1). Different binary words represent different land surface functioning and therefore allow the delineation of functional units (with a focus on the radiation-driven conditions) in the (Attert) catchment. Based on results from Figs. 9 and 12, larger units can be found within the forests (e.g., “00000”, “10000”, “00001”), main settlements or frequently bare soils (”11111”) and large pastures (”11011” and “00100”). The heterogeneous areas are more related to periodical land cover changes and represent small-scale dominations of processes throughout the time series.