On the spatial scaling of soil moisture
Introduction
Soil moisture is a key variable in hydrologic processes at the land surface. It has a major influence on a wide spectrum of hydrological processes including flooding, erosion, solute transport, and land–atmosphere interactions (Georgakakos, 1996). Soil moisture is highly variable in space and knowledge of the characteristics of that variability is important for understanding and predicting the above processes at a range of scales.
There are two main sources of data for capturing the spatial variability of soil moisture. These are remote sensing data and field measurements. While remote sensing data potentially give spatial patterns, interpretation of the remotely sensed signal is often difficult. Specifically, there are a number of confounding factors such as vegetation characteristics and soil texture that may affect the remotely sensed signal much more strongly than the actual soil moisture (De Troch et al., 1996). Also, remote sensing signals give some sort of average value over an area (which is termed the footprint) and it is difficult to relate the soil moisture variability at the scale of the footprint to larger scale or smaller scale soil moisture variability (Stewart et al., 1996). A further complication when interpreting soil moisture patterns obtained from microwave images is that the depth of penetration is poorly defined and can vary over the image. This means that the depth over which the soil moisture has been integrated is unknown and may vary. An alternative is to use field data. However, field data are always point samples and, again, it is difficult to relate the point values to areal averages. Also, field data are often collected in small catchments, while soil moisture predictions are needed in large catchments; and the samples are often widely spaced while, ideally, closely spaced samples are needed.
With both remote sensing and field data, these difficulties arise because the scale at which the data are collected is different from the scale at which the predictions are needed. In other words, the difficulty is related to the need for a “change of scale” from the measurements to the predictive model. This change of scale has been discussed by Blöschl (1998) and Beckie (1996), within the conceptual framework of scaling. Upscaling refers to increasing the scale and downscaling refers to decreasing the scale. Blöschl (1998) noted that the variability apparent in the data will be different from the true natural variability and that the difference will be a function of the scale of the measurements. Similarly, the variability apparent in the parameters or state variables of a model will be different from the true natural variability (and from the variability in the data), and this difference will be a function of the scale of the model.
Blöschl and Sivapalan (1995) suggested that both the measurement scale and the modelling scale consist of a scale triplet consisting of spacing, extent, and support (Fig. 1). ‘Spacing’ refers to the distance between samples or model elements; ‘extent’ refers to the overall coverage; and ‘support’ refers to the integration volume or area. All three components of the scale triplet are needed to uniquely specify the space dimensions of a measurement or a model. For example, for a transect of TDR (time domain reflectometry) soil moisture samples in a research catchment, the scale triplet may have typical values of, say, 10 m spacing (between the samples), 200 m extent (i.e. the length of the transect), and 10 cm support (the diameter of the region of influence of the TDR measurement). Similarly, for a finite difference model of spatial hydrologic processes in the same catchment, the scale triplet may have typical values of, say 50 m spacing (between the model nodes), 1000 m extent (i.e. the length of the model domain), and 50 m support (the size of the model elements or cells).
Blöschl and Sivapalan (1995) and Blöschl (1998) noted that the effect of spacing, extent, and support can be thought of as a filter and should always be viewed as relative to the scale of the natural variability. The scale of the natural variability is also termed the ‘process scale’ and relates to whether the natural variability is small-scale variability or large-scale variability. More technically, the correlation length or the integral scale of a natural process (Journel and Huijbregts, 1978) can quantify the scale of the natural variability (the process scale). The correlation length can be derived from the variogram or the spatial covariance function of the data.
The variogram characterises spatial variance as a function of the separation (lag) of the data points. The main structural parameters of the variogram are the sill and correlation length. The sill is the level at which the variogram flattens out. If a sill exists, the process is stationary and the sill can be thought of as the variance of two distantly separated points. The correlation length is a measure of the spatial continuity of the variable of interest. For an exponential variogram, the correlation length relates to the average distance of correlation. The spatial correlation scale is sometimes characterised by the range instead of the correlation length. The range is the maximum distance of which spatial correlations are present. While the correlation length and the range contain very similar information, the numerical value of the range is three times the correlation length for an exponential variogram.
As mentioned previously, the spatial variability apparent in the data will be different from the true spatial variability. Here, we are interested in the bias in the statistical properties of the true spatial variability, estimated from the measured data. First, the apparent variance in the data will, as a rule, be biased as compared to the true variance, and this bias is a function of the ratio of measurement scale and process scale. If the support of the measurement scale is large as compared with the process scale (the true correlation length), most of the variability will be averaged out and the apparent variance will be smaller than the true variance. This is consistent with the general observation that aggregation always removes variance. However, if the extent is small as compared to the process scale, the large-scale variance is not sampled and the apparent variance will be smaller than the true variance. The bias associated with extent and support is depicted schematically in Fig. 2. Second, the apparent correlation length (or apparent integral scale) in the data will, as a rule, be biased as compared with the true correlation length. Again, this bias is a function of the ratio of measurement scale and process scale. It is clear that large-scale measurements can only sample large-scale variability and small-scale measurements can only sample small-scale variability. As a consequence of this, large measurement scales (in terms of spacing, extent and support), compared to the process scale, will generally lead to apparent correlation lengths that are larger than the true correlation lengths, and small measurement scales will cause an underestimation of the correlation lengths (Fig. 2). The effect of the modelling scale (in terms of spacing, extent and support) will be similar to that of the measurement scale.
While, conceptually, it is straightforward to assess bias related to measurement scale, it may be difficult to estimate it quantitatively. One approach is to use a geostatistical framework (Journel and Huijbregts, 1978, Isaaks and Srivastava, 1989, Gelhar, 1993, Vanmarcke, 1983;). In geostatistics, the spatial variability is represented by the variogram, which is the lag dependent variance of the natural process. Based on the variogram, there are a number of geostatistical techniques available that allow quantitative estimates of each bias mentioned earlier. For example, for analysing the effect of support, regularisation techniques are given in the literature. All of these techniques hinge on the assumption of the variable under consideration being a spatially correlated random variable. However, for the case of soil moisture this is not necessarily a valid assumption. There is substantial evidence from measurements in the literature that soil moisture is indeed spatially organised (Dunne et al., 1975, Rodrı́guez-Iturbe et al., 1995, Georgakakos, 1996, Schmugge and Jackson, 1996, Western et al. 1998a). For example, soil moisture is often topographically organised with connected bands of high soil moisture in the depression zones of a catchment and near the streams. Soil moisture may also be organised as a consequence of landuse patterns, vegetation patterns, soil patterns, geology and other controls.
This paper has two aims. The first is to examine how the apparent spatial statistical properties of soil moisture (variance and correlation length) change with the measurement scale (in terms of spacing, extent and support). The second is to examine whether standard geostatistical techniques of regularisation and variogram analysis are applicable to organised soil moisture patterns. The main feature of this paper is that we use soil moisture data collected in the field with a very high spatial resolution. Real spatial patterns are used because they often have characteristics that do not conform to the assumptions underlying the standard geostatistical approach, yet standard geostatistics are often applied. Of greatest significance here is the existence of connectivity and nonstationarity. These characteristics introduce uncertainty about the applicability of standard geostatistical tools for scaling spatial fields such as soil moisture. Stationary random fields, which are often assumed in geostatistics, do not have these characteristics. Using real data allows assessment of the robustness of the predictions of standard geostatistical techniques to the presence of spatial organisation (connectivity). We are not aware of any paper that examines the effect of connectivity on scaling. The data we use also allows consideration of a range of scales of two orders of magnitude in the analysis.
Section snippets
Field description and data set
The data used to examine the spatial scaling of soil moisture come from the 10.5 ha Tarrawarra catchment. Tarrawarra is an undulating catchment located on the outskirts of Melbourne, Australia (Fig. 3). It has a temperate climate and the average soil moisture is high during winter and low during summer. Very detailed measurements of the spatial patterns of soil moisture have been made in this catchment. In this paper, data from four soil moisture surveys (Table 1) are used. Two of these surveys
Method of analysis
The analysis in this paper consists of three main steps. The first step is an analysis of the full data set, which gives the “true” variogram. The second step is a resampling analysis in which the (empirical) variance and integral scale are estimated. The third step consists of estimating the (theoretical) variance and integral scale directly from the “true” variogram.
In all three steps, an exponential variogram without a nugget is used. Western et al. (1998b) found that the soil moisture data
Results
Fig. 7 shows the results of the resampling analysis. Measurement scales in terms of spacing, extent and support have different effects on the apparent variance. Spacing does not affect the apparent variance. Increasing the extent causes an increase in the apparent variance while increasing the support causes a decrease in the apparent variance. On the other hand increasing the scale in terms of spacing, extent and support always increases the apparent correlation length. These tendencies are
Discussion
The resampling analysis of the soil moisture data at Tarrawarra indicates that bias in the variance and the correlation length does exist as a consequence of the measurement scale. The general shapes of curves representing bias are as suggested by Blöschl (1998). For the ideal case of very small spacings, very large extents and very small supports, the apparent variance and the apparent correlation length are close to their true values. However, as the spacing increases, the extent decreases or
Conclusions
In this paper the effect on the apparent spatial statistical properties of soil moisture (variance and correlation length) of changes in the measurement scale (in terms of spacing, extent and support) has been examined using a resampling analysis. ‘Spacing'refers to the distance between samples; ‘extent’ refers to the overall coverage; and ‘support’ refers to the integration area. For the ideal case of very small spacings, very large extents and very small supports, the apparent variance and
Acknowledgements
The Tarrawarra catchment is owned by the Cistercian Monks (Tarrawarra) who have provided free access to their land and willing cooperation throughout the project. Funding for the above work was provided by the Australian Research Council (project A39531077), the Cooperative Research Centre for Catchment Hydrology, the Oesterreichische Nationalbank, Vienna (project 5309) and the Australian Department of Industry, Science and Tourism, International Science and Technology Program.
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