On the spatial scaling of soil moisture

Abstract

The spatial scale of soil moisture measurements is often inconsistent with the scale at which soil moisture predictions are needed. Consequently a change of scale (upscaling or downscaling) from the measurements to the predictions or model values is needed. The measurement or model scale can be defined as a scale triplet, consisting of spacing, extent and support. ‘Spacing’ refers to the distance between samples; ‘extent’ refers to the overall coverage; and ‘support’ refers to the area integrated by each sample. The statistical properties that appear in the data, the apparent variance and the apparent correlation length, are as a rule different from their true values because of bias introduced by the measurement scale. In this paper, high-resolution soil moisture data from the 10.5 ha Tarrawarra catchment in south-eastern Australia are analysed to assess this bias quantitatively. For each survey up to 1536 data points in space are used. This allows a change of scale of two orders of magnitude. Apparent variances and apparent correlation lengths are calculated in a resampling analysis. Apparent correlation lengths always increase with increasing spacing, extent or support. The apparent variance increases with increasing extent, decreases with increasing support, and does not change with spacing. All of these sources of bias are a function of the ratio of measurement scale (in terms of spacing, extent and support) and the scale of the natural variability (i.e. the true correlation length or process scale of soil moisture). In a second step this paper examines whether the bias due to spacing, extent and support can be predicted by standard geostatistical techniques of regularisation and variogram analysis. This is done because soil moisture patterns have properties, such as connectivity, that violate the standard assumptions underlying these geostatistical techniques. Therefore, it is necessary to test the robustness of these techniques by application to observed data. The comparison indicates that these techniques are indeed applicable to organised soil moisture fields and that the bias is predicted equally well for organised and random soil moisture patterns. A number of examples are given to demonstrate the implications of these results for hydrologic modelling and sampling design.

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.