**Scale Dependence and Time Stability of Nonstationary Soil Water Storage in a Hummocky Landscape Using Global Wavelet Coherency**

Asim Biswas and Bing Cheng Si

Additional information is available at the end of the chapter

http://dx.doi.org/10. 5772/52340

## **1. Introduction**

[35] Si BC, Kachanoski RG, Reynolds WD (2007) Analysis of soil variability. In:E.G. Gre‐

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[39] Vieira SR (2000) Geostatistics in studies of spatial variability of soil. In: Novais, R.F. et al. (Eds.) Topics in soil science. Viçosa, Brazilian soc. soil sci. Vol 1, 1-54 p.

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[41] Webster R, McBratney AB (1989) On the Akaike Information Criterion for choosing

[42] Wilding LP, Bouma J, Goss D (1994) Impact of spatial variability on modeling. In: Bryant, R., Arnold, R.W., (Eds.) Quantitative modelling of soil forming processes.

[43] Yost RS, Uehara G, Fox RL (1982) Geostatistical analysis of soil chemical properties of

[44] Zawadzaki J, Fabijanczyk P (2007) Use of variogramsfro field magnetory analysis in

[45] Zeleke TB, Si BC (2005) Scaling relationships between saturated hydraulic conductiv‐

SSSA Special Publication #39. Soil Sci. Soc. Am. Inc. Madison, WI. 61-75 p.

large land areas. I. Semivariograms. Soil sci. soc.am. j. 46, 1028-1032.

upper Silesia industrial area. Stud. geophys. geod. 51, 535-550.

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of soil properties.Adv. agro. 38, 45-94

96 Advances in Agrophysical Research

Soil water is one of the most important limiting factors for semi-arid agricultural production and a key element in environmental health. It controls a large number of surface and subsurface hydrological processes that are critical in understanding a broad variety of natural processes (geomorphological, climatic, ecological) acting over a range of spatio-temporal scales (Entin et al. , 2000). Knowledge on the behavior of soil water storage and its spatiotemporal distribution provides essential information on various hydrologic, climatic, and general circulation models (Beven, 2001; Western et al. , 2002), weather prediction, evapo‐ transpiration and runoff (Famigleitti and Wood, 1995), precipitation (Koster et al. , 2004) and atmospheric variability (Delworth and Manabe, 1993).

The distribution of soil water in the landscape is the response of a number of highly het‐ erogeneous factors and processes acting in different intensities over a variety of scales (Goovaerts, 1998; Entin et al. , 2000). The heterogeneity in factors and processes make the spatial distribution of soil water highly heterogeneous in space and time and create a challenge in hydrology (Quinn, 2004). Therefore, a large number of samples are needed in order to characterize the field averaged soil water with certain level of precision. Howev‐ er, if a field or watershed is repeatedly surveyed for soil water, some sites or points are consistently wetter or consistently drier than the field average. Vachaud et al. (1985) were the first to examine the similarity of the spatial pattern in soil water storage over time and termed this phenomenon time stability. The time stability is defined as a time invariant association between spatial location and classical statistical measures of soil water, most

© 2013 Biswas and Si; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Biswas and Si; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

often the mean (Grayson and Western, 1998). Authors used the Spearman's rank correla‐ tion to explain the similarity in the overall spatial patterns between two measurement ser‐ ies and the cumulative probability function of relative mean differences to examine the rank similarity of the individual locations over time [Vachaud et al. , 1985]. Various au‐ thors have used this concept to examine the similarity between the spatial patterns of soil water storage over a range of investigated area, sampling scheme, sampling depth, inves‐ tigation period and land use (Kachanoski and de Jong, 1988; Grayson and Western, 1998; Hupet and Vanclooster, 2002; Tallon and Si, 2004; Martínez-Fernández and Ceballos, 2005; Starks et al. , 2006; Cosh et al. , 2008; Hu et al. , 2010b). However, information on the simi‐ larity between the spatial patterns of soil water within a season (intra-season), between seasons (inter-season), or within a season of different years (inter-annual) is not very com‐ mon (Biswas and Si, 2011a).

Kachanoski and de Jong (1988) used the spatial coherency analysis to identify the simi‐ larity of the scales of the spatial patterns of soil water distribution over time and named the phenomena temporal persistence. Their study indicated loss of time stability at the scale < 40 m during the recharge period, which was attributed to topography. The spa‐ tial coherency analysis is based on the spectral analysis (Jenkins and Watts, 1968; Kacha‐ noski and de Jong, 1988), which approximates the spatial data series by a sum of sine and cosine functions. Each function has an amplitude and a frequency or period. While the squared amplitude represents the variance contribution, the frequency component can be used to represent the spatial scale of ongoing processes (Webster, 1977; Shumway and Stoffer, 2000; Brillinger, 2001). The spectral analysis or frequency domain analysis is based on the assumption of second order stationarity (i. e. the mean and the variance of the series are finite and constant). However, more often than not, the soil spatial varia‐ tion is nonstationary. Nonstationarity in the spatial distribution of soil water storage was also mentioned by Kachanoski and de Jong (1988). Nonstationarity restricts direct appli‐ cation of spatial coherency analysis to examine the similarity in the spatial patterns of soil water storage at different scales or scale-specific time stability, which calls for a new method.

Wavelet analysis can deal with localized features and thus nonstationarity by partitioning the spatial variations into locations and frequencies (Lark and Webster, 1999; Grinsted et al. , 2004; Si and Farrell, 2004, Yates et al., 2006; Biswas et al. , 2008), therefore providing an op‐ portunity to study the spatial variation in soil water storage at multiple scales. While, the global wavelet analysis can deal with the scale specific variations, the global wavelet coher‐ ency analysis elucidates the scale specific correlation between any two spatial series. There‐ fore, the global wavelet coherency can be used to examine the similarity in the spatial patterns of soil water storage measured at two different times at multiple scales and study the scale-specific time stability. The objective of this study was to examine the scales of time stability of nonstationary soil water storage at different seasons in a hummocky landscape using the global wavelet coherency.

**Figure 1.** Geographic location of the study site at St Denis National Wildlife Area within Prairie Pothole Region of North America along with the 3-dimensional and cross sectional view of the transect and different landform elements along the transect. CX indicates convex, CV indicates concave, CW indicates cultivated wetlands and UW indicates un‐

Scale Dependence and Time Stability of Nonstationary Soil Water Storage in…

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99

Wavelet analysis (Mallat, 1999) is used to divide a spatial series into different frequency components and study each component using a fully scalable window or wavelet. It cal‐ culates localized variations by shifting the standard function (mother wavelet) along the spatial series. The detail theory of the wavelet analysis is available elsewhere (Farge, 1992; Kumar and Foufoula-Georgiou, 1993, 1997; Torrence and Compo, 1998) and is beyond the scope of this chapter. There are different types of wavelet transform including discrete

cultivated wetlands

**2. Theory**

**Figure 1.** Geographic location of the study site at St Denis National Wildlife Area within Prairie Pothole Region of North America along with the 3-dimensional and cross sectional view of the transect and different landform elements along the transect. CX indicates convex, CV indicates concave, CW indicates cultivated wetlands and UW indicates un‐ cultivated wetlands

## **2. Theory**

often the mean (Grayson and Western, 1998). Authors used the Spearman's rank correla‐ tion to explain the similarity in the overall spatial patterns between two measurement ser‐ ies and the cumulative probability function of relative mean differences to examine the rank similarity of the individual locations over time [Vachaud et al. , 1985]. Various au‐ thors have used this concept to examine the similarity between the spatial patterns of soil water storage over a range of investigated area, sampling scheme, sampling depth, inves‐ tigation period and land use (Kachanoski and de Jong, 1988; Grayson and Western, 1998; Hupet and Vanclooster, 2002; Tallon and Si, 2004; Martínez-Fernández and Ceballos, 2005; Starks et al. , 2006; Cosh et al. , 2008; Hu et al. , 2010b). However, information on the simi‐ larity between the spatial patterns of soil water within a season (intra-season), between seasons (inter-season), or within a season of different years (inter-annual) is not very com‐

Kachanoski and de Jong (1988) used the spatial coherency analysis to identify the simi‐ larity of the scales of the spatial patterns of soil water distribution over time and named the phenomena temporal persistence. Their study indicated loss of time stability at the scale < 40 m during the recharge period, which was attributed to topography. The spa‐ tial coherency analysis is based on the spectral analysis (Jenkins and Watts, 1968; Kacha‐ noski and de Jong, 1988), which approximates the spatial data series by a sum of sine and cosine functions. Each function has an amplitude and a frequency or period. While the squared amplitude represents the variance contribution, the frequency component can be used to represent the spatial scale of ongoing processes (Webster, 1977; Shumway and Stoffer, 2000; Brillinger, 2001). The spectral analysis or frequency domain analysis is based on the assumption of second order stationarity (i. e. the mean and the variance of the series are finite and constant). However, more often than not, the soil spatial varia‐ tion is nonstationary. Nonstationarity in the spatial distribution of soil water storage was also mentioned by Kachanoski and de Jong (1988). Nonstationarity restricts direct appli‐ cation of spatial coherency analysis to examine the similarity in the spatial patterns of soil water storage at different scales or scale-specific time stability, which calls for a new

Wavelet analysis can deal with localized features and thus nonstationarity by partitioning the spatial variations into locations and frequencies (Lark and Webster, 1999; Grinsted et al. , 2004; Si and Farrell, 2004, Yates et al., 2006; Biswas et al. , 2008), therefore providing an op‐ portunity to study the spatial variation in soil water storage at multiple scales. While, the global wavelet analysis can deal with the scale specific variations, the global wavelet coher‐ ency analysis elucidates the scale specific correlation between any two spatial series. There‐ fore, the global wavelet coherency can be used to examine the similarity in the spatial patterns of soil water storage measured at two different times at multiple scales and study the scale-specific time stability. The objective of this study was to examine the scales of time stability of nonstationary soil water storage at different seasons in a hummocky landscape

mon (Biswas and Si, 2011a).

98 Advances in Agrophysical Research

method.

using the global wavelet coherency.

Wavelet analysis (Mallat, 1999) is used to divide a spatial series into different frequency components and study each component using a fully scalable window or wavelet. It cal‐ culates localized variations by shifting the standard function (mother wavelet) along the spatial series. The detail theory of the wavelet analysis is available elsewhere (Farge, 1992; Kumar and Foufoula-Georgiou, 1993, 1997; Torrence and Compo, 1998) and is beyond the scope of this chapter. There are different types of wavelet transform including discrete wavelet transform (DWT), continuous wavelet transform (CWT), wavelet packet trans‐ form (WPT), maximal overlap discrete wavelet transform (MODWT). These are suite of tools and can be used for certain purposes with some advantages and disadvantages. In this study, we use the continuous wavelet transform (CWT), where the wavelet coeffi‐ cients at consecutive scales and locations can carry common information and provide a re‐ dundant representation of the signals information content and thus the detailed scale information (Farge, 1992; Lau and Weng, 1995; Keitt and Fischer, 2006; Furon et al., 2008). The detailed theory of the CWT can be found in various text books including Mallat (1998) and Chui (1992). Briefly, the CWT for a spatial series (*Yi* ) of length *N* (*i*= 1, 2, …,*N*) with an equal sampling interval of *δx*, can be defined as the convolution of *Yi* with the scaled (*s*) and translated (*x*) wavelet (Torrence and Compo, 1998). Wavelet coefficients, *Wi <sup>Y</sup>* (*s*) can be calculated as

$$\mathcal{W}\_{l}^{Y}\left(\mathbf{s}\right) = \sqrt{\frac{\delta\mathbf{x}}{s}} \sum\_{j=1}^{N} \mathbf{Y}\_{j} \boldsymbol{\nu} \left[ \left( j - i \right) \frac{\delta\mathbf{x}}{s} \right] \tag{1}$$

*YZ Y Z W WW i ii*

( ) ( ) <sup>1</sup> 0 1 *<sup>N</sup> YZ YZ*

*n n Ws Ws N* - =

While the global wavelet cross spectra are similar to the covariances in the spatial domain, the global wavelet coherency spectra are similar to the coefficients of determination in the spatial domain for two variables. The global wavelet coherency spectra can be calculated as

( ( )) ( ( ))

where, *S* is the smoothing operator, *S*(*W*¯*YZ* (*s*)) is the smoothed global cross wavelet spectra of spatial series *Y* and *Z*, *S*(*W*¯*<sup>Y</sup>* (*s*)) and *S*(*W*¯*<sup>Z</sup>* (*s*)) are the smoothed global wavelet spectra of the spatial series *Y* and *Z*, respectively. In calculating wavelet coherency, it is necessary to smooth global cross wavelet spectra beforehand; otherwise, it will always be equal to 1 (Torrence and Compo, 1998; Maraun and Kurths, 2004). The coherency should be calculated on expected values. However, in most cases, there is only one realization of a spatial series, thus a coheren‐ cy value has only one degree of freedom. By smoothing the coherency, one can overcome this problem and increase the degrees of freedom. In this study, we have used a boxcar window of size 5 (5 sample point average) to smooth the global wavelet and cross wavelet spectra.

to 1 and measure the correlation between two spatial series at each scale or within a particu‐ lar frequency band. The closer the coherency values to one, the more similar the spatial pat‐

The significance test for the wavelet coherency spectra can be carried out by calculating the confidence interval from an assumed theoretical distribution (Koopmans, 1974). However,

> ( ) ( ) ( ) <sup>2</sup> 2, 2 2, 2 1 *N N*

*n F*

( ) ( ) <sup>1</sup> <sup>2</sup> *R s* =- - 1 1 a

*F*

= 0 vs. *R*<sup>2</sup>

a


a

*SW s SW s*

*YZ Y Z SW s*

2

( ) ( ( ))

Like the coefficient of determination, the global wavelet coherency spectra (*R*<sup>2</sup>

terns at a particular frequency or scale (= sampling interval / frequency).

*R s*

the cutoff points for the test of hypothesis *R*<sup>2</sup>

*F* distribution (Koopmans, 1974):

or

2

*R s*

*<sup>Z</sup>* <sup>∗</sup>is the complex conjugate of *Wi*

where, *Wi*

lated as

\* = (5)

http://dx.doi.org/10. 5772/52340

101

Scale Dependence and Time Stability of Nonstationary Soil Water Storage in…

*<sup>Z</sup>* . The global cross wavelet spectra can be calcu‐

<sup>=</sup> å (6)

= (7)

) range from 0

> 0 can be conducted for *s* ≠ 0 from the

<sup>=</sup> - + (8)

<sup>2</sup>*<sup>m</sup>* (9)

where *ψ*[ ] denotes wavelet function. Out of many wavelet functions, the Morlet wavelet was used in this study because of enhanced spatial and frequency resolution. Morlet wave‐ let can be represented as (Torrence and Compo, 1998)

$$
\mu\left(\eta\right) = \pi^{-1/4} e^{\left(m\eta - 0.5\eta\right)^2} \tag{2}
$$

where, *i* is the complex number and equal to −1, *ω* is the dimensionless frequency and *η* is the dimensionless space. The imaginary part conserved in the wavelet transform with Mor‐ let wavelet can be used to identify the dominant orientation of variations in a random field. The energy associated with a scale and location can be measured from the magnitude of the wavelet coefficient. The wavelet power spectrum can be defined as |*Wi <sup>Y</sup>* (*s*)| <sup>2</sup> , which is the space-frequency-energy representation of a spatial series. The global wavelet spectrum is the average of local wavelet spectra over all locations and is given by,

$$\left| \tilde{\boldsymbol{W}}^{Y} \left( s \right) = \frac{1}{N} \sum\_{n=0}^{N-1} \left| \boldsymbol{W}\_{n}^{Y} \left( s \right) \right|^{2} \tag{3}$$

Similarly, the global wavelet spectra of another spatial series *Z* will be

$$\overline{\mathcal{W}}^{Z}\left(s\right) = \frac{1}{N} \sum\_{n=0}^{N-1} \left| \mathcal{W}\_{\mu}^{Z}\left(s\right) \right|^{2} \tag{4}$$

The cross wavelet spectra between two spatial series *Y* and *Z* can be calculated as

$$\mathcal{W}\_l^{YZ} = \mathcal{W}\_l^Y \mathcal{W}\_l^{Z\*} \tag{5}$$

where, *Wi <sup>Z</sup>* <sup>∗</sup>is the complex conjugate of *Wi <sup>Z</sup>* . The global cross wavelet spectra can be calcu‐ lated as

$$\left| \overline{\mathcal{W}}^{YZ} \left( s \right) = \frac{1}{N} \sum\_{n=0}^{N-1} \left| \mathcal{W}\_n^{YZ} \left( s \right) \right| \tag{6}$$

While the global wavelet cross spectra are similar to the covariances in the spatial domain, the global wavelet coherency spectra are similar to the coefficients of determination in the spatial domain for two variables. The global wavelet coherency spectra can be calculated as

$$R^2(s) = \frac{\left\| \mathbb{S}(\overline{W}^{\times Z}(s)) \right\|^2}{\mathbb{S}\left(\overline{W}^{\times}(s)\right)\mathbb{S}\left(\overline{W}^Z(s)\right)}\tag{7}$$

where, *S* is the smoothing operator, *S*(*W*¯*YZ* (*s*)) is the smoothed global cross wavelet spectra of spatial series *Y* and *Z*, *S*(*W*¯*<sup>Y</sup>* (*s*)) and *S*(*W*¯*<sup>Z</sup>* (*s*)) are the smoothed global wavelet spectra of the spatial series *Y* and *Z*, respectively. In calculating wavelet coherency, it is necessary to smooth global cross wavelet spectra beforehand; otherwise, it will always be equal to 1 (Torrence and Compo, 1998; Maraun and Kurths, 2004). The coherency should be calculated on expected values. However, in most cases, there is only one realization of a spatial series, thus a coheren‐ cy value has only one degree of freedom. By smoothing the coherency, one can overcome this problem and increase the degrees of freedom. In this study, we have used a boxcar window of size 5 (5 sample point average) to smooth the global wavelet and cross wavelet spectra.

Like the coefficient of determination, the global wavelet coherency spectra (*R*<sup>2</sup> ) range from 0 to 1 and measure the correlation between two spatial series at each scale or within a particu‐ lar frequency band. The closer the coherency values to one, the more similar the spatial pat‐ terns at a particular frequency or scale (= sampling interval / frequency).

The significance test for the wavelet coherency spectra can be carried out by calculating the confidence interval from an assumed theoretical distribution (Koopmans, 1974). However, the cutoff points for the test of hypothesis *R*<sup>2</sup> = 0 vs. *R*<sup>2</sup> > 0 can be conducted for *s* ≠ 0 from the *F* distribution (Koopmans, 1974):

$$R^2\left(s\right) = \frac{F\_{2,N-2}\left(a\right)}{n - 1 + F\_{2,N-2}\left(a\right)}\tag{8}$$

or

wavelet transform (DWT), continuous wavelet transform (CWT), wavelet packet trans‐ form (WPT), maximal overlap discrete wavelet transform (MODWT). These are suite of tools and can be used for certain purposes with some advantages and disadvantages. In this study, we use the continuous wavelet transform (CWT), where the wavelet coeffi‐ cients at consecutive scales and locations can carry common information and provide a re‐ dundant representation of the signals information content and thus the detailed scale information (Farge, 1992; Lau and Weng, 1995; Keitt and Fischer, 2006; Furon et al., 2008). The detailed theory of the CWT can be found in various text books including Mallat

with an equal sampling interval of *δx*, can be defined as the convolution of *Yi* with the scaled (*s*) and translated (*x*) wavelet (Torrence and Compo, 1998). Wavelet coefficients,

( )

 d

ë û <sup>å</sup> (1)


<sup>=</sup> å (3)

<sup>=</sup> å (4)

*<sup>Y</sup>* (*s*)| <sup>2</sup>

, which is the

1

<sup>2</sup> 1/4 0.5 ( ) *<sup>i</sup> e* vh h

= é ù <sup>=</sup> - ê ú

y

where *ψ*[ ] denotes wavelet function. Out of many wavelet functions, the Morlet wavelet was used in this study because of enhanced spatial and frequency resolution. Morlet wave‐

where, *i* is the complex number and equal to −1, *ω* is the dimensionless frequency and *η* is the dimensionless space. The imaginary part conserved in the wavelet transform with Mor‐ let wavelet can be used to identify the dominant orientation of variations in a random field. The energy associated with a scale and location can be measured from the magnitude of the

space-frequency-energy representation of a spatial series. The global wavelet spectrum is the

( ) ( ) <sup>1</sup> <sup>2</sup> 0 1 *<sup>N</sup> Y Y*

( ) ( ) <sup>1</sup> <sup>2</sup> 0 1 *<sup>N</sup> Z Z*

The cross wavelet spectra between two spatial series *Y* and *Z* can be calculated as

*n n Ws Ws N* - =

*n n Ws Ws N* - =

) of length *N* (*i*= 1, 2, …,*N*)

(1998) and Chui (1992). Briefly, the CWT for a spatial series (*Yi*

( ) *<sup>N</sup> <sup>Y</sup> i j j x x Ws Y ji s s* d

> yh p

wavelet coefficient. The wavelet power spectrum can be defined as |*Wi*

average of local wavelet spectra over all locations and is given by,

Similarly, the global wavelet spectra of another spatial series *Z* will be

let can be represented as (Torrence and Compo, 1998)

*Wi*

*<sup>Y</sup>* (*s*) can be calculated as

100 Advances in Agrophysical Research

$$\mathcal{R}^2\left(s\right) = 1 - \left(1 - \alpha\right)^{\frac{1}{2m}}\tag{9}$$

where, *α* is the significance level and 2*m* + 1 is the width of the boxcar window. Therefore, *m* is the number of terms in each symmetrical half of the boxcar window. If the calculated co‐ herency*R* ^ 2(*s*) is greater than the theoretical value*<sup>R</sup>* 2(*s*) at a particular scale (*s*), then the cal‐ culated coherency is significantly different from zero at the specified *α*. In this study, we have used *m* = 2, therefore the cutoff point at *α* = 0. 99 is 0. 684.

#### **3. Materials and methods**

A field experiment was conducted at St Denis National Wildlife Area, (52° 12′ N latitude, 106° 50′ W longitude), which is approximately 40 km east of Saskatoon, Saskatchewan, Cana‐ da (Fig. 1). A detailed information on the study site, soil water measurement and calibration of measurement instruments can be found in Biswas et al. (2012) and Biswas and Si (2011a, b). Briefly, the landscape of the study site is hummocky with a complex sequence of slopes (10 to 15%) extending from different sized rounded depression to complex knolls and knobs (Pennock, 2005) and is typical of the Prairie Pothole Region of North America (Fig. 1). The dominant soil type of the study site is Dark Brown Chernozem (Mollosiol in USDA soil taxonomy), which is developed from moderately fine to fine textured, calcareous, glacio-la‐ custrine deposits and modified glacial till (Saskatchewan Centre for Soil Research, 1989). The climate of the study area is semi-arid with the mean annual air temperature of 2o C and the mean annual precipitation of 360 mm, of which 84 mm occurs as snow during winter (AES, 1997). The annual precipitation of the site during 2006, 2007, 2008, and 2009 were 489 mm, 366 mm, 331 mm, and 402 mm, respectively (Fig. 2). Year 2010 received 645 mm rainfall only during the spring and summer months (April to September), which is almost double the long-term average annual precipitation (Environment Canada, 2011).

**Figure 3.** Site specific neutron probe calibration equation completed over three year time (2007-09). P indicates the

Scale Dependence and Time Stability of Nonstationary Soil Water Storage in…

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103

Soil water storage was measured along a transect of 576 m long with equally spaced 128 points (4. 5 m sampling interval). The transect was established over several knolls and sea‐ sonal depressions representing different landform cycles (Fig. 1). Topographic survey of the site was completed using Light Detection and Ranging (LiDAR) survey of the study area at 5 m ground resolution. Different landform elements were also identified as convex (CX), concave (CV), cultivated wetlands (CW) and uncultivated wetland (UW) (Fig. 1). The vege‐ tation of the study site was mixed grass including *Agropyronelongatum*, *Agropyronintermedi‐ um*, *Bromusbiebersteinii*, *Elymusdauricus*, *Festucarubra*, *Onobrychisviciifolia*, *Elymuscanadensis*, *Agropyrontrachycaulum*and *Medicago sativa*, which was seeded in 2004 and allowed to grow every year. Surface 0-20 cm soil water was measured using time domain reflectometry (TDR) probe and a metallic cable tester (Model 1502B, Tektronix, Beaverton, OR, USA). A neutron probe (Model CPN 501 DR Depthprobe, CPN International Inc. , Martinez, CA, USA) was used to measure the soil water down to 140 cm at 20 cm vertical intervals. Soil cores at selected locations within 1 m around the neutron access tube were taken at different moisture conditions and the soil water content of each 10 cm interval were determined by gravimetric methods. The volumetric water content (gravimetric water content × bulk densi‐ ty) and the neutron counts were used to calibrate the neutron probe. The resulting calibra‐

neutron count to standard neutron count) was used to convert the neutron probe count ratio to volumetric soil water at different depths and different locations (Fig. 3). Because neutron probe is prone to error for surface soil water measurements, the average soil water content at the surface 20-cm layer was measured using vertically installed time domain reflectome‐

= 0. 86, where *P* is the ratio of

ratio of the actual neutron count to the standard neutron count.

tion equation (*θv* = 0. 8523 *P* + 0. 0612 with *n* = 101 and *r*<sup>2</sup>

**Figure 2.** Monthly distribution of total precipitation in the year of 2006-09 along with the long term normal (90 year average)

where, *α* is the significance level and 2*m* + 1 is the width of the boxcar window. Therefore, *m* is the number of terms in each symmetrical half of the boxcar window. If the calculated co‐

culated coherency is significantly different from zero at the specified *α*. In this study, we

50′ W longitude), which is approximately 40 km east of Saskatoon, Saskatchewan, Cana‐ da (Fig. 1). A detailed information on the study site, soil water measurement and calibration of measurement instruments can be found in Biswas et al. (2012) and Biswas and Si (2011a, b). Briefly, the landscape of the study site is hummocky with a complex sequence of slopes (10 to 15%) extending from different sized rounded depression to complex knolls and knobs (Pennock, 2005) and is typical of the Prairie Pothole Region of North America (Fig. 1). The dominant soil type of the study site is Dark Brown Chernozem (Mollosiol in USDA soil taxonomy), which is developed from moderately fine to fine textured, calcareous, glacio-la‐ custrine deposits and modified glacial till (Saskatchewan Centre for Soil Research, 1989). The climate of the study area is semi-arid with the mean annual air temperature of 2o

the mean annual precipitation of 360 mm, of which 84 mm occurs as snow during winter (AES, 1997). The annual precipitation of the site during 2006, 2007, 2008, and 2009 were 489 mm, 366 mm, 331 mm, and 402 mm, respectively (Fig. 2). Year 2010 received 645 mm rainfall only during the spring and summer months (April to September), which is almost double

> **Months Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sept. Oct. Nov. Dec.**

**Figure 2.** Monthly distribution of total precipitation in the year of 2006-09 along with the long term normal (90 year

have used *m* = 2, therefore the cutoff point at *α* = 0. 99 is 0. 684.

A field experiment was conducted at St Denis National Wildlife Area, (52°

the long-term average annual precipitation (Environment Canada, 2011).

**2006 (489) 2007 (366) 2008 (331) 2009 (402) 90 yrs Normal (360)**

^ 2(*s*) is greater than the theoretical value*<sup>R</sup>* 2(*s*) at a particular scale (*s*), then the cal‐

12′ N latitude,

C and

herency*R*

102 Advances in Agrophysical Research

106°

**3. Materials and methods**

**Precipitation (mm)**

average)

**0**

**35**

**70**

**105**

**140**

**Figure 3.** Site specific neutron probe calibration equation completed over three year time (2007-09). P indicates the ratio of the actual neutron count to the standard neutron count.

Soil water storage was measured along a transect of 576 m long with equally spaced 128 points (4. 5 m sampling interval). The transect was established over several knolls and sea‐ sonal depressions representing different landform cycles (Fig. 1). Topographic survey of the site was completed using Light Detection and Ranging (LiDAR) survey of the study area at 5 m ground resolution. Different landform elements were also identified as convex (CX), concave (CV), cultivated wetlands (CW) and uncultivated wetland (UW) (Fig. 1). The vege‐ tation of the study site was mixed grass including *Agropyronelongatum*, *Agropyronintermedi‐ um*, *Bromusbiebersteinii*, *Elymusdauricus*, *Festucarubra*, *Onobrychisviciifolia*, *Elymuscanadensis*, *Agropyrontrachycaulum*and *Medicago sativa*, which was seeded in 2004 and allowed to grow every year. Surface 0-20 cm soil water was measured using time domain reflectometry (TDR) probe and a metallic cable tester (Model 1502B, Tektronix, Beaverton, OR, USA). A neutron probe (Model CPN 501 DR Depthprobe, CPN International Inc. , Martinez, CA, USA) was used to measure the soil water down to 140 cm at 20 cm vertical intervals. Soil cores at selected locations within 1 m around the neutron access tube were taken at different moisture conditions and the soil water content of each 10 cm interval were determined by gravimetric methods. The volumetric water content (gravimetric water content × bulk densi‐ ty) and the neutron counts were used to calibrate the neutron probe. The resulting calibra‐ tion equation (*θv* = 0. 8523 *P* + 0. 0612 with *n* = 101 and *r*<sup>2</sup> = 0. 86, where *P* is the ratio of neutron count to standard neutron count) was used to convert the neutron probe count ratio to volumetric soil water at different depths and different locations (Fig. 3). Because neutron probe is prone to error for surface soil water measurements, the average soil water content at the surface 20-cm layer was measured using vertically installed time domain reflectome‐ try probe and a metallic cable tester (Model 1502B, Tektronix, Beaverton, OR, USA). A standard calibration equation (*θ<sup>v</sup>* =0.115 *ka* −0.176, where *ka* = *L* <sup>2</sup> / *L* <sup>2</sup> is the dielectric con‐ stant, *L*2 is the distance between the arrival of signal reflected from the probe-to-soil inter‐ face and the signal reflected from the end of the probe curves (measured from waveform) and *L* is the length of the TDR probe) following Topp and Reynolds (1998) was used to de‐ rive the water content from the TDR recordings. Soil water content was measured 25 times at different seasons during a year over a five-year period (2007 – 2011). Based on the season, the measurements were divided into three groups: spring, summer, and fall. Though the analysis was completed for all the measurements in all the years, the space restriction in this chapter and the for demonstration purposes, only the result from 2008 and 2009 were used. The results from these years were very similar to the results from other years and can be generalized over the measurement period.

with the increase in time between measurements indicating the decrease in the degree of time stability over time. For example, the rank correlation coefficient was 0. 97 between the measurement series on 20 April 2009 and 7 May 2009, which gradually decreased to 0. 82

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**Figure 4.** Spatial distribution of selected soil water storage series along the transect. The value in *italics* presents the

average soil-water storage.

between the measurement series on 20 April 2009 and 27 October 2009.

Wavelet analysis was completed using the MATLAB (The MathWorks Inc. ) code written by Torrence and Compo (1998) and is available online at http://paos. colorado. edu/research/ wavelets/. The graphs were prepared in SigmaPlot (Systat Software Inc. ).

## **4. Results**

High water storage was found at the locations of 100 to 140 m and 225 to 250 m from the origin of the transect, which were situated within depressions (Fig. 4). On contrary, the knolls stored less water. However, the difference of stored water in depressions and on knolls reduced from spring to fall within a year. For example, the range was 43. 42 cm for 20 April 2009 during spring and was 20. 81 cm for 27 October 2009 during fall season (Fig. 4). Apparently, the mean and variance of the first half of transect was quite different from that of the second half. It is evident that soil water along the transect is non-stationary.

Spearman's rank correlation coefficients were used to examine the similarity of the overall spatial pattern. High rank correlation coefficients between any two-measurement series in‐ dicated time stability of overall spatial pattern of soil water storage. There was very strong intra-season time stability. For example, the rank correlation coefficient was 0. 98 between the measurement series on 2 May 2008 and 31 May 2008 (spring) and was 0. 99 between the measurement series on 23 August 2008 and 17 September 2008 (summer). Similarly, there was also strong inter-annual time stability. For example, the rank correlation coefficient was 0. 96 between the measurement series on 2 May 2008 (spring) and 7 May 2009 (spring) and was 0. 97 between the measurement series on 22 October 2008 (fall) and 27 October 2009 (fall). However, a relatively low rank correlation coefficient was observed between the measurement series from two different seasons. For example, the rank correlation coefficient was 0. 89 between the measurement series on 31 May 2008 (spring) and 23 August 2008 (summer), and 0. 85 between the measurement series on 31 May 2008 (spring) and 22 Octo‐ ber 2008 (fall). However, the correlation coefficient was 0. 99 between the measurement ser‐ ies on 23 August 2008 (summer) and 22 October 2008 (fall) indicating strong similarity between summer and fall measurements. The correlation coefficients gradually decreased with the increase in time between measurements indicating the decrease in the degree of time stability over time. For example, the rank correlation coefficient was 0. 97 between the measurement series on 20 April 2009 and 7 May 2009, which gradually decreased to 0. 82 between the measurement series on 20 April 2009 and 27 October 2009.

try probe and a metallic cable tester (Model 1502B, Tektronix, Beaverton, OR, USA). A standard calibration equation (*θ<sup>v</sup>* =0.115 *ka* −0.176, where *ka* = *L* <sup>2</sup> / *L* <sup>2</sup> is the dielectric con‐ stant, *L*2 is the distance between the arrival of signal reflected from the probe-to-soil inter‐ face and the signal reflected from the end of the probe curves (measured from waveform) and *L* is the length of the TDR probe) following Topp and Reynolds (1998) was used to de‐ rive the water content from the TDR recordings. Soil water content was measured 25 times at different seasons during a year over a five-year period (2007 – 2011). Based on the season, the measurements were divided into three groups: spring, summer, and fall. Though the analysis was completed for all the measurements in all the years, the space restriction in this chapter and the for demonstration purposes, only the result from 2008 and 2009 were used. The results from these years were very similar to the results from other years and can be

Wavelet analysis was completed using the MATLAB (The MathWorks Inc. ) code written by Torrence and Compo (1998) and is available online at http://paos. colorado. edu/research/

High water storage was found at the locations of 100 to 140 m and 225 to 250 m from the origin of the transect, which were situated within depressions (Fig. 4). On contrary, the knolls stored less water. However, the difference of stored water in depressions and on knolls reduced from spring to fall within a year. For example, the range was 43. 42 cm for 20 April 2009 during spring and was 20. 81 cm for 27 October 2009 during fall season (Fig. 4). Apparently, the mean and variance of the first half of transect was quite different from that

Spearman's rank correlation coefficients were used to examine the similarity of the overall spatial pattern. High rank correlation coefficients between any two-measurement series in‐ dicated time stability of overall spatial pattern of soil water storage. There was very strong intra-season time stability. For example, the rank correlation coefficient was 0. 98 between the measurement series on 2 May 2008 and 31 May 2008 (spring) and was 0. 99 between the measurement series on 23 August 2008 and 17 September 2008 (summer). Similarly, there was also strong inter-annual time stability. For example, the rank correlation coefficient was 0. 96 between the measurement series on 2 May 2008 (spring) and 7 May 2009 (spring) and was 0. 97 between the measurement series on 22 October 2008 (fall) and 27 October 2009 (fall). However, a relatively low rank correlation coefficient was observed between the measurement series from two different seasons. For example, the rank correlation coefficient was 0. 89 between the measurement series on 31 May 2008 (spring) and 23 August 2008 (summer), and 0. 85 between the measurement series on 31 May 2008 (spring) and 22 Octo‐ ber 2008 (fall). However, the correlation coefficient was 0. 99 between the measurement ser‐ ies on 23 August 2008 (summer) and 22 October 2008 (fall) indicating strong similarity between summer and fall measurements. The correlation coefficients gradually decreased

of the second half. It is evident that soil water along the transect is non-stationary.

wavelets/. The graphs were prepared in SigmaPlot (Systat Software Inc. ).

generalized over the measurement period.

**4. Results**

104 Advances in Agrophysical Research

**Figure 4.** Spatial distribution of selected soil water storage series along the transect. The value in *italics* presents the average soil-water storage.

Because of the nonstationarity of soil water along the transect, the scale specific similarity in the spatial pattern of soil water storage was examined using global wavelet coherency. There was strong intra-season time stability during summer or fall compared to spring (Fig. 5). Statistically significant strong coherency at all scales during summer and fall indicated that the spatial patterns present at different scales in summer were also present in fall. Simi‐ larly, large significant coherency between the measurement series of spring 2008 and spring 2009 or summer 2008 and summer 2009 indicated strong inter-annual time stability (Fig. 6). However, non-significant coherency at the scales < 20 m in the spring indicated the loss of intra-season time stability (Fig. 5). Similarly, there was loss of inter-annual time stability be‐ tween springs of different years at the scales < 30 m (Fig. 6). However, the time stability was lost at scales < 65 m between the spring and summer and < 70 m between the spring and fall measurement series (inter-season) (Fig. 7). There was strong time stability between the summer and fall at all scales (Fig. 7). The minimum scale of statistically significant coheren‐ cy was the lowest within a season and gradually increased with the increase in time be‐ tween measurements (Fig. 8). For example, the minimum scale of significant coherency was 25 m between the measurement series on 20 April 2009 and 7 May 2009 and was 40 m be‐ tween the measurement series on 20 April 2009 and 27 May 2009. The correlation between the measurement series on 20 April 2009 and 27 October 2009 was not significant at almost

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**Scale (m) 0 50 100 150 200**

**Figure 7.** Global wavelet coherency spectra of inter-season (spring-summer, spring-fall, and summer-fall of 2008) time

**31 May 2008 - 23 Aug 2008 (0.89)**

**31 May 2008 - 22 Oct. 2008 (0.85)**

**23 Aug. 2008 - 22 Oct. 2008 (0.99)**

**[spring - summer]**

**[spring - fall]**

**[summe - fall]**

**99 % Confidence level**

all scales except from 80 to 120 m (Fig. 8).

**Global wavelet coherency (***R*

**0.0**

stability. Values in the parentheses are the rank correlation coefficients.

**0.2**

**0.4**

**0.6**

**0.8**

**1.0**

*2*

**)**

**Figure 5.** Global wavelet coherency spectra of intra-season (spring, summer and fall of 2008) time stability. Values in the parentheses are the rank correlation coefficients.

**Figure 6.** Global wavelet coherency spectra of inter-annual (spring, summer, and fall of 2008 and 2009) time stability. Values in the parentheses are the rank correlation coefficients.

Because of the nonstationarity of soil water along the transect, the scale specific similarity in the spatial pattern of soil water storage was examined using global wavelet coherency. There was strong intra-season time stability during summer or fall compared to spring (Fig. 5). Statistically significant strong coherency at all scales during summer and fall indicated that the spatial patterns present at different scales in summer were also present in fall. Simi‐ larly, large significant coherency between the measurement series of spring 2008 and spring 2009 or summer 2008 and summer 2009 indicated strong inter-annual time stability (Fig. 6). However, non-significant coherency at the scales < 20 m in the spring indicated the loss of intra-season time stability (Fig. 5). Similarly, there was loss of inter-annual time stability be‐ tween springs of different years at the scales < 30 m (Fig. 6). However, the time stability was lost at scales < 65 m between the spring and summer and < 70 m between the spring and fall measurement series (inter-season) (Fig. 7). There was strong time stability between the summer and fall at all scales (Fig. 7). The minimum scale of statistically significant coheren‐ cy was the lowest within a season and gradually increased with the increase in time be‐ tween measurements (Fig. 8). For example, the minimum scale of significant coherency was 25 m between the measurement series on 20 April 2009 and 7 May 2009 and was 40 m be‐ tween the measurement series on 20 April 2009 and 27 May 2009. The correlation between the measurement series on 20 April 2009 and 27 October 2009 was not significant at almost all scales except from 80 to 120 m (Fig. 8).

**Scale (m) 0 50 100 150 200**

**Scale (m) 0 50 100 150 200**

**99 % Confidence level**

**2 May 2008 - 7 May 2009 (0.97)**

**23 Aug. 2008 - 27 Aug. 2009 (0.96)**

**22 Oct. 2008 - 27 Oct. 2009 (0.96)**

**99 % Confidence level**

**[spring]**

**[summer]**

**[fall]**

**Figure 5.** Global wavelet coherency spectra of intra-season (spring, summer and fall of 2008) time stability. Values in

**[spring]**

**[summer]**

**Figure 6.** Global wavelet coherency spectra of inter-annual (spring, summer, and fall of 2008 and 2009) time stability.

**[fall]**

**31 May 2008 - 21 June 2008 (0.99)**

**16 July 2008 - 23 Aug. 2008 (0.96)**

**17 Sept. 2008 - 22 Oct. 2008 (0.99)**

**Global wavelet coherency (***R*

**Global wavelet coherency (***R*

**0.0**

Values in the parentheses are the rank correlation coefficients.

**0.2**

**0.4**

**0.6**

**0.8**

**1.0**

*2*

**)** **0.0**

the parentheses are the rank correlation coefficients.

**0.2**

**0.4**

**0.6**

**0.8**

**1.0**

*2*

**)**

106 Advances in Agrophysical Research

**Figure 7.** Global wavelet coherency spectra of inter-season (spring-summer, spring-fall, and summer-fall of 2008) time stability. Values in the parentheses are the rank correlation coefficients.

## **5. Discussions**

In our study area, depressions receive snowmelt runoff water from the surrounding uplands and store more water compared to knolls during the spring (Gray et al. , 1985; Winter and Rosenberry, 1995). In addition, uneven distribution of drifting snow in the landscape also contributes to the high water storage in depressions (Woo and Rowsell, 1993; Hayashi et al. , 1998; Lungal, 2009). Therefore, the alternate knolls and depressions along the transect creat‐ ed a spatial pattern in soil water storage inverse to the spatial pattern of elevation (Fig. 4). This spatial pattern with topography persisted through summer until fall (Fig. 4) as the de‐ pressions always stored more water than knolls. However, the variable demand of evapo‐ transpiration reduced the difference in the maximum and minimum soil water storage (range) over time within a year. This phenomenon repeated every year.

face roughness, soil texture). Therefore, there was a loss of intra-season time stability at small scales during the spring. The loss of time stability at small scales was also observed between the spatial series from the spring of different years (Fig. 6). However, the influence of these small-scale processes was not strong enough to change the overall spatial pattern. Therefore, a strong intra-season and inter-annual time stability was observed in the overall spatial patterns (identified from Spearman's rank correlation coefficients) irrespective of

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109

However, with the establishment of vegetation, runoff from rainfall became the rare event and therefore the macro-topography was not able to reestablish the spatial patterns weak‐ ened by differential water uptake by vegetation (e. g. , more water uptake in depression than on knolls). Though macro-topography is still important, it is the interactions between vegeta‐ tion and macro-topography that determined the spatial patterns of soil water. Because the in‐ teractions were relatively similar in later summer and early fall, the spatial patterns of soil water storage were very similar in summer and fall at all scales (Fig. 5). The intra-season and inter-annual time stability of the overall and the scale specific spatial patterns at all seasons contradicted the results of various authors who mentioned that the wet season showed strong time stability (Gómez-Plaza et al. , 2000; Qiu et al. , 2001; Hupet and Vanclooster, 2002), while others pointed out to the dry season (Robinson and Dean, 1993; Famiglietti et al. , 1998).

In addition, the processes controlling the spatial pattern of soil water storage may not change abruptly. For example, the vegetation growth is gradual and so is the evapotranspi‐ ration demand. In our study, large coherency between the measurement series on 20 April 2009 and 7 May 2009 indicated strong intra-season time stability. However, with the in‐ crease in time difference between measurements relatively weaker inter-season time stabili‐ ty was resulted (Fig. 8). This result contradicted the findings of Grayson et al. (1997), who indicated that the local controls (e. g. , vegetation, soil texture) are dominant in dry season (evapotranspiration > precipitation), while nonlocal controls (e. g. , topography) dominate redistribution of water during wet period or moisture surplus conditions (evapotranspira‐ tion < precipitation). However, the consistent coherency with reduced magnitude in our study indicated the change in the degree of the same control, but not a switch to different

This study was different from Kachanoski and de Jong (1988), who used spatial coherency analysis to identify the scale specific similarity of the spatial patterns of soil water storage. The spatial coherency analysis assumed that the data series was stationary. However, au‐ thors identified the nonstationary nature of soil water storage (Kachanoski and de Jong, 1988) and divided the transect to create piecewise stationary series. Conversely, the global wavelet coherency analysis was able to deal with nonstationary soil water series. The wave‐ let analysis is well established to deal with nonstationarity. In addition, the conclusion of Kachanoski and de Jong (1988) was based on one-year measurement of soil water, which may not be universal as the precipitation variability over years may create a different experi‐ mental situation in different years. In this study, we have confirmed our conclusion based

on five years of measurement of soil water storage.

controls.

moisture conditions in different seasons (Martínez-Fernández and Ceballos, 2003).

**Figure 8.** Global wavelet coherency spectra between the soil water measurement on 20 April 2009 and the measure‐ ments at different time within the year 2009. Values in the parentheses are the rank correlation coefficients.

At the absence of vegetation during spring, the soil water is lost mainly through surface evaporation, and to a lesser extent, the ground water interaction (Hayashi et al. , 1998; van der Kamp et al. , 2003). Evaporation may be higher in south-facing slope than in north-fac‐ ing slope, but the difference in evaporation due to aspects may not be able to diminish the spatial patterns of soil water storage due to nonlocal controls (e. g. , macro-topography: knolls and depression) at large scales (Kachanoski and de Jong, 1988; Grayson et al. , 1997). However, the spatial patterns created from the micro-topography were not strong enough to dominate the differential evaporation created from the difference in local controls (e. g. , sur‐

face roughness, soil texture). Therefore, there was a loss of intra-season time stability at small scales during the spring. The loss of time stability at small scales was also observed between the spatial series from the spring of different years (Fig. 6). However, the influence of these small-scale processes was not strong enough to change the overall spatial pattern. Therefore, a strong intra-season and inter-annual time stability was observed in the overall spatial patterns (identified from Spearman's rank correlation coefficients) irrespective of moisture conditions in different seasons (Martínez-Fernández and Ceballos, 2003).

**5. Discussions**

108 Advances in Agrophysical Research

**Global wavelet coherency (***R*

**0.0**

**0.2**

**0.4**

**0.6**

**0.8**

**1.0**

*2*

**)**

In our study area, depressions receive snowmelt runoff water from the surrounding uplands and store more water compared to knolls during the spring (Gray et al. , 1985; Winter and Rosenberry, 1995). In addition, uneven distribution of drifting snow in the landscape also contributes to the high water storage in depressions (Woo and Rowsell, 1993; Hayashi et al. , 1998; Lungal, 2009). Therefore, the alternate knolls and depressions along the transect creat‐ ed a spatial pattern in soil water storage inverse to the spatial pattern of elevation (Fig. 4). This spatial pattern with topography persisted through summer until fall (Fig. 4) as the de‐ pressions always stored more water than knolls. However, the variable demand of evapo‐ transpiration reduced the difference in the maximum and minimum soil water storage

> **Scale (m) 0 50 100 150 200**

**Figure 8.** Global wavelet coherency spectra between the soil water measurement on 20 April 2009 and the measure‐

At the absence of vegetation during spring, the soil water is lost mainly through surface evaporation, and to a lesser extent, the ground water interaction (Hayashi et al. , 1998; van der Kamp et al. , 2003). Evaporation may be higher in south-facing slope than in north-fac‐ ing slope, but the difference in evaporation due to aspects may not be able to diminish the spatial patterns of soil water storage due to nonlocal controls (e. g. , macro-topography: knolls and depression) at large scales (Kachanoski and de Jong, 1988; Grayson et al. , 1997). However, the spatial patterns created from the micro-topography were not strong enough to dominate the differential evaporation created from the difference in local controls (e. g. , sur‐

ments at different time within the year 2009. Values in the parentheses are the rank correlation coefficients.

**7 May 2009 (0.97) 27 May 2009 (0.96) 21 July 2009 (0.90) 27 August 2009 (0.88) 27 October 2009 (0.82) 99% Confidence level**

(range) over time within a year. This phenomenon repeated every year.

However, with the establishment of vegetation, runoff from rainfall became the rare event and therefore the macro-topography was not able to reestablish the spatial patterns weak‐ ened by differential water uptake by vegetation (e. g. , more water uptake in depression than on knolls). Though macro-topography is still important, it is the interactions between vegeta‐ tion and macro-topography that determined the spatial patterns of soil water. Because the in‐ teractions were relatively similar in later summer and early fall, the spatial patterns of soil water storage were very similar in summer and fall at all scales (Fig. 5). The intra-season and inter-annual time stability of the overall and the scale specific spatial patterns at all seasons contradicted the results of various authors who mentioned that the wet season showed strong time stability (Gómez-Plaza et al. , 2000; Qiu et al. , 2001; Hupet and Vanclooster, 2002), while others pointed out to the dry season (Robinson and Dean, 1993; Famiglietti et al. , 1998).

In addition, the processes controlling the spatial pattern of soil water storage may not change abruptly. For example, the vegetation growth is gradual and so is the evapotranspi‐ ration demand. In our study, large coherency between the measurement series on 20 April 2009 and 7 May 2009 indicated strong intra-season time stability. However, with the in‐ crease in time difference between measurements relatively weaker inter-season time stabili‐ ty was resulted (Fig. 8). This result contradicted the findings of Grayson et al. (1997), who indicated that the local controls (e. g. , vegetation, soil texture) are dominant in dry season (evapotranspiration > precipitation), while nonlocal controls (e. g. , topography) dominate redistribution of water during wet period or moisture surplus conditions (evapotranspira‐ tion < precipitation). However, the consistent coherency with reduced magnitude in our study indicated the change in the degree of the same control, but not a switch to different controls.

This study was different from Kachanoski and de Jong (1988), who used spatial coherency analysis to identify the scale specific similarity of the spatial patterns of soil water storage. The spatial coherency analysis assumed that the data series was stationary. However, au‐ thors identified the nonstationary nature of soil water storage (Kachanoski and de Jong, 1988) and divided the transect to create piecewise stationary series. Conversely, the global wavelet coherency analysis was able to deal with nonstationary soil water series. The wave‐ let analysis is well established to deal with nonstationarity. In addition, the conclusion of Kachanoski and de Jong (1988) was based on one-year measurement of soil water, which may not be universal as the precipitation variability over years may create a different experi‐ mental situation in different years. In this study, we have confirmed our conclusion based on five years of measurement of soil water storage.

Time stability is a result of multiple factors. Due to difference in the intensity of different factors, time stability of soil water and its scale dependence can be different. Instead of over-generalizing time stability, we classified time stability into intra-season, inter-season, and inter-annual time stability, because of the similar intra-season and inter-annual hydro‐ logical processes, but different inter-season hydrological processes. Therefore, the conclu‐ sion of this study may lead to improved prediction of soil water from reduced number of monitoring sites, thus allowing improved runoff and stream flow prediction in scarcely gauged basins.

**Author details**

**References**

Asim Biswas1,2\* and Bing Cheng Si2

earth sys. sci. 5: 1–12.

Philadelphia, PA. 540 p.

water resour. 16: 3–20.

res. 105: 11865–11877.

\*Address all correspondence to: asim.biswas@mcgill.ca

hummocky landscape. J hydrol. 408: 100-112.

Can. j. soil sci. (in Press), 92, 649-663.

1 Department of Natural Resource Sciences, McGill University, Canada

2 Department of Soil Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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[1] AES (1997) Canadian daily climate data for Western Canada. In [CD-ROM]. Atmos‐

[2] Beven K (2001) How far can we go in distributed hydrological modeling? Hydrol.

[3] Biswas A Si, BC (2011a). Scale and locations of time stability of soil water storage in a

[4] Biswas A, Chau HW, Bedard-Haughn AK, Si BC (2012) Factors controlling soil water storage in the Hummocky landscape of the prairie pothole region of North America.

[5] Biswas A, Si BC (2011b) Identifying scale specific controls of soil water storage in a hummocky landscape using wavelet coherency analysis. Geoderma165: 50-59.

[6] Biswas A, Si BC (2011c) Revealing the controls of soil water storage at different scales

[7] Biswas A, Si BC, Walley FL (2008) Spatial relationship between δ15N and elevation

[8] Brillinger, D. R. (2001). Time series: Data analysis and theory. Soc. Ind. Appl. Math.

[9] Chui, C. K. (1992). An introduction to wavelets. Academic Press, New York, NY.

[10] Cosh MH, Jackson TJ, Moran S, Bindlish R (2008) Temporal persistence and stability of surface soil moisture in a semi-arid watershed. Remote sens. environ. 112: 304–313.

[11] Delworth T, Manabe S (1993) Climate variability and land-surface processes. Adv.

[12] Entin JK, Robock A, Vinnikov KY, Hollinger SE, Liu S, Namkhai A (2000) Temporal and spatial scales of observed soil moisture variations in the extra tropics. J. geophys.

in a Hummocky landscape. Soil sci. soc. am. j. 75: 1295–1306.

in agricultural landscapes. Nonlinear proc. geoph. 15: 397–407.

pheric Environment Service, Environment Canada, Downsview, ON.

## **6. Summary and conclusions**

The similarity in the overall spatial pattern of soil water storage was first examined by Va‐ chaud et al. (1985) and termed as the time stability of the spatial pattern. Kachanoski and de Jong (1988) extended the concept of time stability to the scale dependence of time stability using spatial coherency analysis. However, the stationarity assumption of the spatial coher‐ ency analysis restricts the use of this method for nonstationary spatial series. We have used global wavelet coherency analysis to examine the scale dependence of intra-season, interseason and inter-annual time stability of nonstationary soil water spatial patterns.

There was strong intra-season time stability of the overall and scale specific spatial pattern. The time stability was lost at the scales < 20 m within the spring and < 30 m between the spring measurements from different years. However, strong time stability was present at all scales during the summer and fall, when the high evapotranspiration demand created simi‐ lar spatial patterns. Similar processes in the summer and fall resulted strong inter-season time stability. However, not so similar processes in spring created weaker inter-season time stability between the spring and summer or the spring and fall. There was loss of time sta‐ bility at the scales < 65 m and < 70 m between the spring and summer and the spring and fall, respectively. However, the change in the scales of time stability was not abrupt; rather it gradually decreased with the increase of time difference between measurements. The change in the similarity of the spatial patterns of soil water storage over time at different scales is an indicative of the change in the hydrological processes operating at those scales. Therefore, the analysis outcome can be used to identify the change in the sampling domain as controlled by the hydrological processes operating at different scales delivering the maxi‐ mum information with minimum sampling effort.

## **Acknowledgments**

The project was funded by a CSIRO Post Doctoral Fellowship and the University of Sas‐ katchewan. Authors appreciate help from Amanda, Danny, Jason, Khizir and Henry in field data collection.

## **Author details**

Time stability is a result of multiple factors. Due to difference in the intensity of different factors, time stability of soil water and its scale dependence can be different. Instead of over-generalizing time stability, we classified time stability into intra-season, inter-season, and inter-annual time stability, because of the similar intra-season and inter-annual hydro‐ logical processes, but different inter-season hydrological processes. Therefore, the conclu‐ sion of this study may lead to improved prediction of soil water from reduced number of monitoring sites, thus allowing improved runoff and stream flow prediction in scarcely

The similarity in the overall spatial pattern of soil water storage was first examined by Va‐ chaud et al. (1985) and termed as the time stability of the spatial pattern. Kachanoski and de Jong (1988) extended the concept of time stability to the scale dependence of time stability using spatial coherency analysis. However, the stationarity assumption of the spatial coher‐ ency analysis restricts the use of this method for nonstationary spatial series. We have used global wavelet coherency analysis to examine the scale dependence of intra-season, inter-

There was strong intra-season time stability of the overall and scale specific spatial pattern. The time stability was lost at the scales < 20 m within the spring and < 30 m between the spring measurements from different years. However, strong time stability was present at all scales during the summer and fall, when the high evapotranspiration demand created simi‐ lar spatial patterns. Similar processes in the summer and fall resulted strong inter-season time stability. However, not so similar processes in spring created weaker inter-season time stability between the spring and summer or the spring and fall. There was loss of time sta‐ bility at the scales < 65 m and < 70 m between the spring and summer and the spring and fall, respectively. However, the change in the scales of time stability was not abrupt; rather it gradually decreased with the increase of time difference between measurements. The change in the similarity of the spatial patterns of soil water storage over time at different scales is an indicative of the change in the hydrological processes operating at those scales. Therefore, the analysis outcome can be used to identify the change in the sampling domain as controlled by the hydrological processes operating at different scales delivering the maxi‐

The project was funded by a CSIRO Post Doctoral Fellowship and the University of Sas‐ katchewan. Authors appreciate help from Amanda, Danny, Jason, Khizir and Henry in field

season and inter-annual time stability of nonstationary soil water spatial patterns.

gauged basins.

110 Advances in Agrophysical Research

**6. Summary and conclusions**

mum information with minimum sampling effort.

**Acknowledgments**

data collection.

Asim Biswas1,2\* and Bing Cheng Si2

\*Address all correspondence to: asim.biswas@mcgill.ca

1 Department of Natural Resource Sciences, McGill University, Canada

2 Department of Soil Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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**Chapter 6**

**Improving Soil Primary Productivity Conditions with**

As stated by the editors Brandt, C. and Thornes, J., 1996, "Both the vegetation and land use in the Mediterranean areas of Europe strongly reflect human activity since at least the Bronze Age". Putting it briefly for the recent past *ca*. ninety-year period, the natural land‐ scape covered by the original "silvalusitana" has been deforested for agricultural purposes and substituted by monocultures of cereal crops, mainly wheat and barley, aiming at trans‐ forming the overall Alentejo region, southern continental Portugal, into the "granary of Por‐

The so transformed natural landscape resulted in what has been called cereal steppe or pseudo-steppe, since the 1930´s, when the "wheat campaign" started (Pedroso, M. et al., 2009). However, such a "new" land-use system soon revealed to be aggressive to the envi‐ ronment and of discussable sustainability for the soil physics, chemistry and ecology. Nega‐ tive impacts to the landscape physiognomy and dynamics and also soil quality have been declared when it has been realized that the initial and abundant soil organic matter (SOM) content (say then, the original (accumulated) soil's capital") was inexorably exhausted due to its induced unbalance. In fact, the regional, Mediterranean climate is prone to accelerate the SOM mineralization process in combinations with the intensification of the crop produc‐

tion system through, for example, the action of mechanical soil ploughing practices.

The persistence in space and time of such un intensive rain-fed, land use system for cereals production has let to the lessening of (1) crops dry matter (DM) production per hectare (*ha*), or yield; (2) rain-water use efficiency (WUE); (3) the light/radiation use efficiency (LUE) and (4) the overall efficiency of the energy-input into the agro-ecosystem per unit crop DM pro‐

> © 2013 Sampaio and C. Lima; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Sampaio and C. Lima; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Minimum Energy Input in the Mediterranean**

Elsa Sampaio and Júlio C. Lima

http://dx.doi.org/10.5772/52466

tugal" (Pedroso, M. et al., 2009).

**1. Introduction**

Additional information is available at the end of the chapter

