Spatiotemporal behavior of soil water is essential to understand the science of hydrodynamics. Data intensive measurement of surface soil water using remote sensing has established that the spatial variability of soil water can be described using the principle of self-similarity (scaling properties) or fractal theory. This information can be used in determining land management practices provided the surface scaling properties are kept at deep layers. The current study examined the scaling properties of sub-surface soil water and their relationship to surface soil water, thereby serving as supporting information for plant root and vadose zone models. Soil water storage (SWS) down to 1.4 m depth at seven equal intervals was measured along a transect of 576 m for 5 years in Saskatchewan. The surface SWS showed multifractal nature only during the wet period (from snowmelt until mid- to late June) indicating the need for multiple scaling indices in transferring soil water variability information over multiple scales. However, with increasing depth, the SWS became monofractal in nature indicating the need for a single scaling index to upscale/downscale soil water variability information. In contrast, all soil layers during the dry period (from late June to the end of the growing season in early November) were monofractal in nature, probably resulting from the high evapotranspirative demand of the growing vegetation that surpassed other effects. This strong similarity between the scaling properties at the surface layer and deep layers provides the possibility of inferring about the whole profile soil water dynamics using the scaling properties of the easy-to-measure surface SWS data.
Knowledge on the spatial distribution of soil water over a range of spatial scales and time has important hydrologic applications including assessment of land–atmosphere interactions (Sivapalan, 1992), performance of various engineered covers, monitoring soil water balance, and validating various climatic and hydrological models (Rodriguez-Iturbe et al., 1995; Koster et al., 2004). However, high variability in soil is a major challenge in hydrology (Quinn, 2004) as the distribution of soil water in the landscape is controlled by various factors and processes operating at different intensities over a variety of extents (Entin et al., 2000). The individual and/or combined influence of these physical factors (e.g., topography, soil properties) and environmental processes (e.g., runoff, evapotranspiration, and snowmelt) gives rise to complex and nested effects, which in turn evolve a signature in the spatial organization (Western et al., 1999) or patterns in soil water as a function of spatial scale (Kachanoski and de Jong, 1988; Kim and Barros, 2002; Biswas and Si, 2011a). This complexity makes the management decision difficult at a scale other than that of measurement. Therefore, it is necessary to transfer variability information from one extent (e.g., pedon) to another (e.g., large catchment), which is called scaling.
The scaling of soil water is possible if the distribution of some statistical parameters (e.g., variance) remain similar at all studied scales. This feature, known as scale-invariance, means that the spatial feature in the distribution of soil water will not change if the length scales are multiplied by a common factor (Hu et al., 1997). Generally, the soil water will have a typical size or scale, a value around which individual measurements are centered. So the probability of measuring a particular value will vary inversely as a power of that value, which is known as the power-law decay, a typical principle of the scaling process. Now, as the spatial distribution of soil water follows the power-law decay (Hu et al., 1997; Kim and Barros, 2002; Mascaro et al., 2010), the spatial variability can be investigated and characterized quantitatively over a large range of measurement extents using the fractal theory (Mandelbrot, 1982). When the spatial distribution of soil water is the response of some linear processes, the scaling can be done using a single coefficient over multiple scales and the distribution shows monofractal behavior. However, the spatial distribution of soil water is the nonlinear response of multiple factors and processes acting over a variety of scales and therefore needs multiple scaling indices (multifractals) for quantifying spatial variability (Hu et al., 1997; Kim and Barros, 2002; Mascaro et al., 2010).
The multifractal behavior in the surface soil water as a result of temporal evolution of wetting and drying cycles has been reported from the sub-humid environment of Oklahoma by Kim and Barros (2002). Mascaro et al. (2010) reported the multifractal behavior of soil water, which was ascribed as a signature of the rainfall spatial variability. Though these measurements can provide a quick estimate of soil water over a large area, they are limited to very few centimeters of the soil profile. These studies reported the multifractal behavior of only the surface soil water indicating the superficial scaling properties. Surface soil layer is exposed to direct environmental forces and is the most dynamic in nature. The scaling properties of surface soil water can be used for land management practices, provided the observed scaling properties remain the same for the deep layers such as vadose zone or the whole soil profile. Understanding the overall hydrological dynamics in soil profiles requires information on the scaling properties and the nature of the spatial variability of soil water over a range of scales at deep layers as well (Biswas et al., 2012c). The information on the similarity in the nature of the spatial variability of soil water between the surface layer and deep layers may also help inferring about the soil profile hydrological dynamics. Therefore, the objectives of this study were to examine over time the scaling properties of sub-surface layers and their relationship with surface layers at different initial soil water conditions. We have examined the scaling properties of soil water storage at each layer and their trend with increasing depth from the surface (cumulative depth) over a 5-year period from a hummocky landscape from central Canada using the multifractal approach. The relationship between the scaling properties of the surface layer and the subsurface layers was also examined using the joint multifractal analysis.
A field experiment was carried out at St. Denis National Wildlife Area
(52
Conceptual schematics showing the vegetation growth patterns over the landscape at different times of the year. The figure is developed based on field observations and the scale is arbitrary.
Various methods including geostatistics (Grego et al., 2006), spectral
analysis (Kachanoski and de Jong, 1988), and wavelet analysis
(Biswas and Si, 2011a, b) have been used to examine the scale-dependent
spatial patterns of SWS. These methods generally deal with how the second
moment of SWS changes with scales or frequencies. When the statistical
distribution of SWS is normal, the second moment plus the average provide a
complete description of the spatial series. However, for other distributions
(e.g., left skewed distribution) higher-order moments are necessary for a
complete description of the spatial series. For example, let us define the
Soil water is highly variable in space and time. If the variability in the spatial/temporal distribution remains statistically similar at all studied scales, the SWS is assumed to be self-similar (Evertsz and Mandelbrot, 1992). Self-similarity, also called scale invariance, is closely associated with the transfer of information from one scale to another. We used the multifractal analysis to explore self-similarity or inherent differences in scaling properties of SWS in this study.
On the spatial domain of the studied field, multifractal analysis was used
to characterize the scaling property of SWS by statistically measuring the
mass distribution (Zeleke and Si, 2004). The spatial domain or the
data along the transect was successively divided into self-similar segments
following the rule of the binomial multiplicative cascade
(Evertsz and Mandelbrot, 1992). This method required that the
two segments divided from a unit interval to be of equal length. With
regards to a unit mass
For SWS spatial series, the scale-invariant mass exponent, was termed as
In a similar manner to Eq. (1), the
Moreover, the fractal dimension of the subsets of segments in scale size
The multifractal spectrum is a powerful tool in portraying the similarity
and/or differences between the scaling properties of the measures
(e.g., SWS). The width of the spectrum (
In addition to the multifractal spectrum, [
While the multifractal analysis characterized the distribution of a SWS
spatial series along its geometric support, the joint multifractal analysis
was used to characterize the joint distribution of two SWS spatial series
along a common geometric support. As an extension of the multifractal
analysis, the length of the data sets was also divided into several segments
of size
Average SWS for the surface 0–20 cm layer over the 5-year period was
5.51 cm. A slight decrease in SWS was observed at the immediate deep layer
(20–40 cm) and a gradual increase thereafter. The 5-year average SWS was 5.45,
5.48, 5.56, 5.61, 5.69, and 5.77 cm for the 20–40, 40–60,
60–80, 80–100, 100–120, and 120–140 cm layers, respectively. Average
SWS for a single measurement varied from 3.40 to 7.16 cm. The highest
average SWS for the surface layer was observed on 29 June 2011. The study
area received a large amount of spring snowmelt (2010 received 642 mm, double
the annual average precipitation) and rainfall during 2011 leading to the
high SWS in the surface layer (Weather Canada historical report). The lowest
average SWS for the surface layer was observed on 23 August 2008, which was
one of the driest summers within the 5-year study period. The highest
average SWS (on 29 June 2011) at the surface layer gradually decreased to
6.55 cm at the deepest layer and the lowest average SWS (on 23 August 2008)
at the surface layer gradually increased to 5.28 cm at the 120–140 cm layer
(Table 1). These top and bottom boundaries formed a wider range (3.76 cm) of
the average SWS at the surface layer compared to that at the deepest layer
(1.27 cm). A big range (2.00 cm) in the standard deviation (maximum
The maximum SWS at the surface layer also varied widely (maximum
The variations in SWS with time were evaluated within a year. There was little change in the average SWS over measurements within the years from 2007 to 2011 except for 2008 (Table 1). For example, average SWS was 6.47, 6.03, 6.54, and 6.33 cm on 6 April 2010, 19 May 2010, 14 June 2010, and 28 September 2010, respectively. However, the average SWS in 2008 drops from 6.28 cm on 2 May 2008 to 3.51 cm on 17 September 2008 in the surface 0–20 cm layer. This falling trend was observed at all soil layers. When compared between years, the trend over time and with depth was very similar in 2007 and 2009 while slightly different between 2010 and 2011 (Table 1). A decreasing trend of the variability was also observed with time. For example, the CV of the surface layer was around 28 % on 2 May 2008, which gradually decreased to around 13 % on 17 September 2008 (Table S1).
Maximum, minimum, and average soil water storage (cm) at different depths (20 cm increment) over the whole measurement period.
Log–log plot between the aggregated variance of the SWS spatial series and the scale. A linear relationship indicated the presence of scale invariance and scaling laws for three selected dates.
The average water storage for soil layers with increasing depth was also calculated by adding the individual layers together. The time-averaged values of SWS were 10.96, 16.44, 22.00, 27.61, 33.30, and 39.07 cm for the 0–40, 0–60, 0–80, 0–100, 0–120, and 0–140 cm, respectively (Table S2). The CV of the 0–20 cm layer was the highest during the wet period and gradually declined to the smallest during the dry period (Table S3). The variability also gradually decreased with depth.
Mass exponents for soil water storage spatial series measured at
selected 20 cm soil layer down to 140 cm in 2008 for a range of
The power-law relationships and the statistical scale invariance were
evaluated using a log–log plot of the aggregated variance of SWS spatial
series at different depths of soil layers and the level of disaggregation
(or scales) at different
The
With the maximum deviation at the surface layer, the
The SSR values gradually decreased and the slopes became almost at unity with
increasing depth (Fig. 4). For example, the SSR values were 14.11, 9.31,
7.71, 6.86, 6.71 and 6.30 and the slopes (single fit) were 0.98, 0.99, 0.99,
1.00, 1.00, and 1.00, respectively, for 0–40, 0–60, 0–80, 0–100, 0–120, and
0–140 cm layer (Table S5). The slopes of the segmented fit
for the
Mass exponents for selected soil water storage spatial series from
surface to different soil layers (cumulative storage) at 20 cm increment
down to 140 cm in 2008 for a range of
The width of the multifractal spectrum (
A decreasing trend in the SSR value was also observed over time within a year. During the dry period, the slopes (single fit and segmented fit) became almost at unity with no significant difference (Table S6). For example, the SSR value was 14.12, 8.25, 1.30, 1.46, and 0.52 and the slope was 0.99, 0.99, 1.00, 1.00, and 1.00, respectively, for the surface layer (0–20 cm) of 21 June 2008, 16 July 2008, 23 August 2008, 17 September 2008 and 22 October 2008 (Fig. 3). Similarly, a small SSR value and consistent slope were also observed at the deepest layer (120–140 cm). The SSR values of the 120–140 cm were 2.47, 2.47, 3.31, 3.44, and 4.57, respectively, for the measurements on 21 June 2008, 16 July 2008, 23 August 2008, 17 September 2008, and 22 October 2008 (Table S6). The slope (single fit) for all these measurements was equal to 1.01 (Fig. 3). There was very little difference in the slopes of the segmented fits.
A significant difference in the slopes of the segmented fit was observed for the surface layer (0–20 cm) of three measurements in 2007 (17 July, 7 August, and 1 September; Fig. S1), and three measurements in 2009 (21 April, 7 May, and 27 May) (Table S4; Fig. S2). The difference became non-significant with depth and during other measurement times. The trend in deep layers over time was very similar to that of 2008. However, the trend in the SSR values and the slopes with time was different in 2010 and 2011 (Table S6). There was very little difference in the SSR values at different times of the year. For example, the SSR value for the surface layer (0–20 cm) was 20.79, 27.18, 24.63, and 26.66 and the slope (single fit) was 0.97, 0.97, 0.97, and 0.97, respectively, for the measurements on 6 April 2010, 19 May 2010, 14 June 2010, and 28 September 2010 (Fig. 3). The slope of the segmented fit of the surface layer (0–20 cm) was significant for all measurements in 2010 and 2011. However, the trend with depth was similar to other years.
The height of the multifractal spectrum at different depths of measurement
was very similar over time. The width of the spectrum (
Multifractal spectra of soil water storage spatial series measured
at each 20 cm soil layer down to 140 cm in 2008, 2010 and 2011 for a range
of
The trend of the
A very similar height of the
Multifractal spectra of soil water storage spatial series from
surface to different soil layers (cumulative storage) at 20 cm increment
down to 140 cm in 2008, 2010, and 2011 for a range of
The information dimension (
Generalized dimension spectra of soil water storage spatial series
measured at each 20 cm soil layer down to 140 cm in 2008 for a range of
Generalized dimension spectra of soil water storage spatial series
from surface to different soil layers (cumulative storage) at 20 cm increment
down to 140 cm in 2008 for a range of
Generally, the
Correlation coefficients between joint multifractal indices (
There were strong correlations between the scaling property of the joint
distribution of the surface soil layer and the deep soil layers. The narrow
width and the diagonally oriented contours between SWS measured on
22 October 2008 at 0–20 and 20–40 cm layers clearly demonstrate strong
association between those two layers (Fig. 11). The correlation between the
surface 0–20 cm and the deep layers on 2 May 2008 (wet period) was larger
than 0.9 (significant at
The amount of water stored in the soil is the result of the dominant underlying hydrological processes. Located in semi-arid climate, the study area receives about 30 % of the long-term annual average precipitation as snowfall during winter months (Pomeroy et al., 2007). Generally, the depressions receive snow from surrounding uplands or knolls as redistributed by strong prairie wind (Pomeroy and Gray, 1995; Fang and Pomeroy, 2009). The snow melts within a short period of time during the early spring and contributes a large amount of water. The frozen ground restricts infiltration and redistributes excess water within the landscape with greater accumulation in depressions (Fig. 1) (Gray et al., 1985). Apart from the snowmelt, the spring rainfall also contributes to the water inflow in the landscape (Fig. 1). This created a spatial pattern of SWS that was almost a mirror image of the spatial distribution of relative elevation (Biswas and Si, 2011a, c; Biswas et al., 2012a).
In the spring, the sources of water loss were the deep drainage and the
evaporation. As the loss of water through deep drainage in the study area
was as low as 2 to 40 mm per year, occurring mainly through the fractures
and preferential flow paths (Hayashi et al., 1998; van der Kamp et al.,
2003), the major loss occurred mainly through evaporation from the surface
of the bare ground and standing water in depressions. These processes lose a
very small amount of water compared to the input of water in spring and
early summer leaving the soil wet. Moreover, the surface soil with high
organic matter content and low bulk density stored a larger amount of water
than the deep layers where the organic matter gradually decreased and the
bulk density increased. Reflecting the long-term history of vegetation
growth in the landscape, the variability of organic matter content
(CV
As the vegetation developed in summer, strong evapotranspiration resulted in the lowest average SWS. High amounts of water in the depressions allowed grasses to grow faster and transpire more water compared to the knolls (Fig. 1). For example, the aquatic vegetation growth within the depressions was as high as 2 m, while the grasses on the knolls grew to a maximum of 1 m tall. The uneven growth of vegetation and the high evapotranspirative demand in summer narrowed the range of SWS. In the soil where water is more available, evapotranspiration will be stronger while the less evapotranspirative demand will be shown in the relatively dry soil. As a result, the excessive water in the relatively wet soil will be offset by evapotranspiration, reducing the disparities between maximum and minimum values. This variable water uptake was visible in the growth of vegetation in the later part of the growing season as well (Fig. 1). The reduction in the range of SWS was the largest in the surface layer and gradually decreased at deeper layers. This is because the surface layer was exposed to various environmental forces. For example, plants can take up more than 70 % of the water they need from the top 50 % of the root zone (Feddes et al., 1978). This dynamic behavior of the surface layer exhausted readily available water and finally reduced the range in water storage. This decrease in range also happened in the later part of the growing season.
Multifractal spectra of joint distribution of SWS at 0–20 and 20–40 cm measured on 22 October 2008. Contour lines show the joint scaling dimensions of the SWS measurement series.
The multifractal and joint multifractal analyses explained the scaling
behavior of SWS at different depths over time. The linearity in the log–log
plot between the aggregated variance in SWS and the scale at all soil layers
over time indicated that SWS behaved under scaling laws (Fig. 2). The near
unity slope of the
Generally during the wet period, excess water fills and drains macropores quickly and creates variations in SWS. Variations in the evaporation due to uneven solar incidence over micro-topography also triggered SWS variability in the surface layer. Additionally, the snowmelt and the release of water controlled by local (e.g., soil texture) and non-local (e.g., topography) factors also affected the spatial distribution of SWS, making it more heterogeneous in the wet period (Grayson et al., 1997; Biswas and Si, 2012). To the contrary, as depth increased, less impact of environmental factors tended to create less variability in SWS and exhibited a monofractal behavior, which was consistent with the uniform slope shown in Fig. 3. During the dry period or later part of the growing season, the SWS storage variability at all depths was small and exhibited monofractal behavior (Fig. 3). Accordingly, the deeper layers in the wet period and all layers in the dry period can be accurately represented by only one scaling exponent while the surface layer in the wet period may require a hierarchy of exponents. A similar trend was observed in SWS of cumulative depth layers (Fig. 4). Resulting from increasing buffering capacity of the deeper soil layers, the variability of cumulative SWS overlaid the multifractal nature of the surface layer, and finally exhibited monofractal behavior in general.
The scaling patterns of SWS at different depths and periods were further
examined using multifractal spectrum [
To sum up, both the unity slope of the
The height of the spectrum,
The two upper soil layers during the wet period tended to exhibit a longer tail of the curve on the left, showing more heterogeneity in the distribution of large values. However, when stepping into the dry period, the spectrum tended to display a longer tail on the right compared to the left side, suggesting more heterogeneity in the distribution of smaller values. A few locations with standing water lead to the spatial differences during the wet period, whereas a few points with very small SWS due to high evapotranspiration by growing vegetation during the dry period results in the heterogenic distribution in smaller values.
The generalized dimension,
Joint multifractal distribution between the surface to various subsurface layers indicated the similarity in the scaling patterns (Table 2). Basically, the hydrological processes of shallower layers were similar to those of the top layer, while deeper layers showed more disparities from the surface. The nearest subsurface (20–40 cm) layer showed generally the highest similarity with the surface (0–20 cm) layer. However, in the wet period, the subsurface layers displayed the smallest similarity to the surface layer, suggesting a higher dynamic nature of hydrological processes. In the dry period, a stronger effect of vegetation overwhelmed the effect of small variations of water distribution, thus creating a more uniform distribution of SWS at all soil layers (Table 2).
Overall, our result revealed a multifractal behavior of surface soil layers during the wet period due to the dynamic nature of hydrological processes. This behavior gradually changed with depth and time (Fig. 12). In the deeper layers during the wet period, the behavior became less multifractal or nearly monofractal. Similarly, in the dry period, the vegetation development and its high evapotranspirative demand in the semi-arid climate of the study area increasingly buffered the variation of SWS, as a result, all the soil layers showed uniform distribution or monofractal behavior (Fig. 12).
Conceptual schematics showing vegetation development over time, dominant water loss processes and the scaling behavior of soil water storage at different depths. The figure is developed based on field observations and scaling analysis. The scale of the figure is arbitrary.
The transformation of information on soil water variability from one scale to another requires knowledge on the scaling behavior and the quantification of scaling indices. Surface soil water can be easily measured (e.g., remote sensing) and presents multi-scaling behavior (requiring multiple scaling indices). However, land-management practices require the understanding of the hydrological dynamics in the root zone and/or the whole soil profile.
In this manuscript, the scaling properties of soil water storage at different soil layers measured over a 5-year period were examined using multifractal and joint multifractal analysis. The scaling properties of soil water storage mainly suggested a monofractal scaling behavior. However, the surface layer in the wet period or with high soil water storage tended to be multifractal, which gradually became monofractal with depth. With the decrease in soil water storage, the scaling behavior became monofractal during the growing season. In a year with high annual precipitation, the soil stored more water in the surface layer throughout the growing period and displayed nearly multifractal scaling behavior. This multifractal nature indicated that the transformation of information from one scale to another at the surface layer during the wet period requires multiple scaling indices. On the contrary, the transformation requires a single scaling index during the dry period for the whole soil profile. The scaling properties of the surface layer were highly correlated with those of the deep layers, which indicated a highly similar scaling behavior in the soil profile. The study was conducted in an undulating landscape from a semi-arid climate and the results were very consistent over the years. Therefore, the observation completed at the field scale in this type of landscape and climate may be generalized in similar landscapes and climatic situations, otherwise may need to be examined thoroughly. The method used here can be transferred to examine the scaling properties in other experimental situations.
The project was funded by the Natural Science and Engineering Research Council of Canada. The help from the graduate student and the summer students of the Department of Soil Science at the University of Saskatchewan in collecting field data is highly appreciated. Edited by: J. M. Miras Avalos Reviewed by: two anonymous referees