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Stochastic modelling of soil water dynamics and sustainability for three vegetation types on the Chinese Loess Plateau.


Soil water acts as an important connection between the atmospheric water, surface water and groundwater components of the water cycle. As such, it plays an important role in the processes of accumulation, transformation and consumption of water resources (Rodriguez-Iturbe 2000; Rodriguez-Iturbe et al. 2001; Ridolfi et al. 2003). In particular, soil water affects the climate, especially the microclimate zones close to the soil and vegetation surfaces, the dynamics of vegetation growth, soil biochemistry and the groundwater regime (Rodriguez-Iturbe et al. 1991; Laio et al. 20016; Porporatoand D'Odorico 2004). Additionally, soil water is the principal limiting factor for plant growth in arid and semiarid ecosystems because it constrains the transpiration demand of plant communities (Hu et al. 2009). Moreover, the degree to which vegetation depends on stored soil water is dependent on the frequency of rain as well as the amount (Milly 1993; Zhuo et al. 2015). Accordingly, an understanding of soil water dynamics at various scales is crucial to studies of ecological and hydrological processes. Therefore, accurate estimations of changes in soil water can provide essential information for the rational management of water resources and land practices, e.g. during vegetation restoration projects (Gao et al. 2013).

Soil water dynamics inevitably have probabilistic and stochastic characteristics due to the large number of involved variables that exhibit uncertainty, which can result from spatial and temporal variability (Rodriguez-Iturbe et al. 1999; Rodriguez-Iturbe 2000; Laio et al. 2001a). These variables include rainfall (amount, frequency and intensity), evapotranspiration, heterogeneity of soil structure, soil texture, topography and the spatial distribution of plant roots both within the soil and across a landscape. Therefore, using a stochastic model to describe soil water dynamics is both logical and useful.

Although the first application of the stochastic concept to soil water balances was reported by Eagleson (1978), a stochastic model that described soil water dynamics in relation to soil water storage was first presented by Milly (1993). However, the model did not consider the dependence of evapotranspiration on soil water, which meant that it could neither be used to evaluate the effect of soil water dynamics on vegetation conditions nor the effect of changes in evapotranspiration on soil water dynamics. An important stochastic model for soil water dynamics was developed later by Rodriguez-Iturbe et al. (1999). This model represented the inputs and outputs of soil water in a physically realistic manner by fully considering the intermittent nature of precipitation and the nonlinear dependence of infiltration, evapotranspiration and leakage on soil water status. The model permitted use of an analytical method that determined the steady-state probability density functions (PDFs) for soil water under various conditions. This approach enabled the roles of climate, soil conditions and vegetation types on soil water at the daily timescale to be assessed by examining the PDF. This method was further improved by Laio et al. (2001 a) by applying two soil water thresholds, the hygroscopic point ([s.sub.e_h]) and the permanent wilting point ([s.sub.e_w]) to the evapotranspiration term, which could more accurately describe soil water dynamics under conditions of water stress, and especially during drought. Subsequent models such as those of Ridolfi et al. (2003) and Laio et al. (2006) analysed the effects of soil topography and vertical root distribution on the stochastic distribution of soil water dynamics.

On the Loess Plateau in China, the 'Grain for Green' project was implemented in 1999 with the objective of reducing soil and water losses by planting trees and converting the land use on steeper slopes from cropland to forest, shrub or grass land (Fu et al. 2006). However, in this region the limited rainfall and deep underground water table makes soil water the primary limiting factor affecting vegetation restoration (Hu et al. 2009; Gao et al. 2013). Soil water replenishment by rainwater is often insufficient to completely recharge soil water consumption (Jian et al. 2015). Consequently, overuse of soil water by vegetation has resulted in soil desiccation and formation of dried soil layers (Chen et al. 2008; Fu et al. 2012). Soil desiccation promotes the development of drought-resistant, deep-rooted trees of 3-5 m of height, which are not efficient in reducing soil erosion (Fu et al. 2012). The soil water dynamics and water balance should be considered when implementing vegetation restoration. Accurate prediction of soil water dynamics is essential in choosing an appropriate type or types of vegetation and optimum vegetation coverage that not only effectively control soil erosion, but also balance soil water consumption with available water resources thereby maintaining sustainable vegetation restoration (McVicar et al. 2010). Therefore, the most sustainable vegetation type is defined as the optimal vegetation coverage which can balance soil water consumption and the eco-environmental service performances (such as effective soil erosion control).

Although the stochastic models of Laio et al. (2001a) and their modified versions (Rodriguez-Iturbe et al. 2007; Laio et al. 2009; Tamea et al. 2009) have been applied to other regions (Laio et al. 2001 b; Porporato et al. 2003; Liu et al. 2007; Pumo et al. 2008; Teuling et al. 2009; Tamea et al. 2010; Huang et al. 2013; Li et al. 2016a), no such study has been conducted on the dynamics and probabilistic simulation of soil water in the presence of deep-rooted plants (Table 1). This study aims to rectify this omission by applying one of these models to soil water dynamics under Loess Plateau conditions. This study aims to (1) analyse the probabilistic characteristics of soil water under three vegetation types on the Loess Plateau, (2) simulate the soil water PDF using the Laio stochastic model (Laio et al. 2001 a, 20016) and (3) determine the most sustainable vegetation types using the simulated soil water PDFs under different coverages. Data for three years of soil water content measurements under three representative land use types (forest, shrub and grass land) on the Loess Plateau were used to accomplish these objectives.

Materials and methods

Study site

The field experiment was conducted in the Wangdonggou watershed (35[degrees]12'N, 107[degrees]42'E and elevation 946-1226 m above mean sea level (AMSL); area 8.3 [km.sup.2]) of the Changwu Agro-ecological Experiment Station, Chinese Academy of Sciences and Ministry of Water Resources. The study watershed is located in the gully region of the Loess Plateau and in the middle reaches of the Yellow River. The region has a continental monsoon climate with a mean temperature of 9.2[degrees]C and a mean open pan evaporation of 1440 mmx[year.sup.-1], based on data collected during 1957-2014 (Duan et al. 2016a). The mean annual precipitation is 578 mm, more than 58% of which falls during July-September. The groundwater is 80 m below the soil surface, which precludes upward capillary flow into the root zone (Liu et al. 2010). The soil texture at the study site is classed as silty clay loam, which is homogeneously distributed in the 80 m vadose zone (Qiao et al. 2018). Dominant plant species include grasses such as Bothriochloa isehaemum L. (BOI), shrubs such as Caragana korshinkii Kom. and sea-buckthorn (Hippophae rhamnoides L.) (SEB), and trees such as black locust (Robinia pseudoacacia) and Chinese pine (Pinus tabulaeformis Carr.) (CHP). The BOI and SEB are deciduous plants, and CHP is evergreen.

Experimental design

In 2003, three experimental plots were established on a natural steep and homogeneous slope (35[degrees]) at an elevation of 1180 m AMSL. The steep slope represents 65% of the watershed area (Duan et al. 20166). To avoid the influences of slope aspect and differences in initial soil properties on the processes of water balance, all plots were established on the same hillslope at the same elevation and, therefore, had the same aspects and similar soil properties. Each plot was 20 m x 5 m with the longest side in the direction of the slope gradient. Each plot was surrounded by a cement wall that stood 15 cm above the ground surface and extended down to a depth of 25 cm, which isolated plot runoff and infiltration. Runoff was funnelled through an outlet at the lower end of each plot where it was collected and measured in a barrel with capacity of ~0.75 [m.sup.3]. Three of the local dominant plant species were each planted in a separate plot- BOI, SEB and CHP representing the land use types grassland, shrub land and forest respectively. No interflow and sub-surface flow appeared during the study period, which implies that most deep percolation was vertical to the water table before moving laterally to rivers in groundwater flow (Zhang et al. 2018). Three 5.5-m access tubes were installed at distances of 5,10 and 15 m from the upper end of each plot along the midline in order to monitor soil water.

Data observations

The soil volumetric water content to a depth of 5 m was measured by a neutron probe (CNC-503B DR, ChaoNeng, China) calibrated using standard methods (Huang and Gallichand 2006; Fu et al. 2012). During April-November in 2011, 2013 and 2014, volumetric soil water content was measured four times per month in the rainy season (July-September) and twice per month in the dry season, which occurred before and after the rainy season. The measurements in 2011 were missed due to neutron probe maintenance. Measurements were made at depth increments of 0.1 and 0.2 m in the 0-1 and 1-5 m soil layers respectively. The mean soil water content in the active root depth profile was determined using depth weighting; 159 sets of data from each experimental plot were collected and used to describe the probability distribution of soil water content. No soil water measurement was conducted during winter due to limited snow with the average snow water equivalent of 30 mm and average annual soil freezing depth of 0.68 m (Huang et al. 2003).

The surface runoff from each experimental plot was measured for each rainfall event that produced small runoff during May-September in the three years.

Daily meteorological parameters, such as rainfall amount and intensity, temperature and wind velocity, were recorded at an automatic weather station in Changwu Agri-ecological Experiment Station at an elevation of 1190 m AMSL, which was 0.5 km from the experimental plots.

The measured volumetric soil water contents were converted to relative soil water content (se) using Eqn 1 (Rodriguez-Iturbe et al. 1999):

[s.sub.e] = [V.sub.w]/[V.sub.a] + [V.sub.w] = [theta]/Po (i)

where [V.sub.w] and [V.sub.a] are the volumes of water and air respectively; [theta] is the measured volumetric water content; and Po is the soil porosity.

Stochastic model

The stochastic model of soil water dynamics used in this study was the one modified by Laio et al. (2001a). The Laio model assumes that (1) soil is modelled as a horizontal layer of the active root depth with homogeneous characteristics, (2) soil moisture is uniform in the vertical direction of root zone and (3) rainfall is a random process and follows Poisson distribution and the evapotranspiration, runoff and so on are determined by analytical solutions. Therefore, the Laio model is only used to describe the long-term statistical characteristics of soil moisture dynamics. The soil water balance at a point for the active root zone is expressed as:

Po [Z.sub.r] d[s.sub.e](t)/dt = [??] [[s.sub.e](t); t] - [chi][[s.sub.e](t);t] (2)

where [Z.sub.r] is the active root depth; se(t) is the relative soil water content (0 < [s.sub.e] (t) < 1); [phi][[s.sub.e](t);t] is the rate of infiltration from rainfall; and [chi][[s.sub.e](t);t] is the rate of soil water loss from the active root depth. The [phi][[s.sub.e](t);t] and [chi][[s.sub.e](t);t] terms can be expressed as follows:

[phi][[s.sub.e](t);t] = R(t) - I(t) - Q[[s.sub.e](t);t] (3)

[chi][[s.sub.e](t);t] = E[[s.sub.e](t);t] + L[[s.sub.e](t);t] (4)

where R(t) is the rainfall rate, I(t) is the amount of rainfall lost through interception by canopy cover, Q[[s.sub.e](t);t] is the rate of surface runoff and E[[s.sub.e](t);t] and L[[s.sub.e](t);t] are the rates of evapotranspiration and leakage respectively.

In this simple representation, the occurrence of rainfall was idealised as a series of point events in continuous time, arising according to a rate ([lambda]), each carrying a random amount of rainfall (h), extracted from an exponential distribution with a mean ([alpha]) as follows:

f(h) = 1/a x [e.sup.(l/[alpha]xh)], [R.sup.2] = 0.995 (5)

Canopy interception is included in the model by considering a threshold value of rainfall depth ([DELTA]), which was assumed to only be related to vegetation type (Laio et al. 2001a). If a single rainfall depth was less than the threshold value, all of the rainfall was intercepted; if the rainfall depth exceeded the threshold value, the difference between the depth and the threshold was the effective precipitation. Four relative soil water content thresholds were used to describe the evapotranspiration process (Fig. 1): the hygroscopic point ([s.sub.e_h]); the permanent wilting point ([s.sub.e_w]), the relative soil water content below which plants begin closing their stomata ([s.sub.e.sup.*]); and the soil field capacity ([s.sub.e_fc]). The term E[[s.sub.e](t);t] incorporated the water losses due to evaporation from the soil and transpiration from the plant. Over the daily timescale the water losses could be defined by three phases: (1) E[[s.sub.e](t)] slowly increased linearly from 0 at [s.sub.e,h] to [E.sub.w] at [s.sub.e,w]; (2) E[[s.sub.e](t)] increased at a relatively moderate rate from [E.sub.w] at [s.sub.e_w] to [E.sub.max] at [s.sub.e.sup.*]; and (3) E[[s.sub.e](t)] occurred under conditions of no evapotranspiration stress, where evapotranspiration remained constant at [E.sub.max] when soil water was between [s.sub.e.sup.*] and [s.sub.e_fc]. When the relative soil water content was greater than [s.sub.e_fc], then leakage L[[s.sub.e](t);t] was dominant, while E[se(t);t] was determined by the potential evapotranspiration (PET) rate (Rodriguez-Iturbe et al. 2001). Assuming no interactions with underlying soil layers and the water table, then Z,[s(t);t] represented vertical percolation with free drainage (Laio et al. 2001a; Rodriguez-Iturbe et al. 2001):

L[[s.sub.e](t),t] = [K.sub.s]/[e.sup.[beta](1-[s.sub.e_fc])] - 1 [[e.sup.[beta](1-[s.sub.e_fc])] - 1] (6)

where [K.sub.s] is the saturated hydraulic conductivity and [beta] = 2i> + 4, where b is a pore size distribution index, and both [beta] and b are empirically fitted parameters related to the shape of the soil water retention curve (Laio et al. 2001 a). All values of [s.sub.e_h], [s.sub.e_w], [s.sub.e.sup.*] and [s.sub.e_fc] are related to the corresponding soil water matric potentials, and [s.sub.e_w] and se* are also related to vegetation type. The [E.sub.max] values are obtained using physically-based expressions, such as the Penman-Monteith equation (Laio et al. 2001a). All model results were interpreted at the daily timescale (Laio et al. 2001a; Rodriguez-Iturbe et al. 2001).

The stochastic rainfall forcing in Eqn 2 makes its solution meaningful only in probabilistic terms. The PDF of soil water, p([s.sub.e]), can be derived from the forward Chapman-Kolmogorov equation for the process taking the limit as t[right arrow][infinity] (Laio et al. 2001 a). The general solution is obtained as:

[mathematical expression not reproducible] (4)

where [eta] is the normalised mean daily evapotranspiration rate under unrestricted soil water conditions; [[eta].sub.w] is the normalised mean daily evaporation rate when [s.sub.e] = [s.sub.e_w]; m is a parameter used in the representation of the leakage loss rate; [gamma] is the inverse of the normalised mean rainfall depth, dimensionless; and [lambda]' is the arrival rate of rainfall events in which the amount of rainfall exceeds the canopy interception.

The input (rainfall) and output (evapotranspiration and leakage losses) of soil water were normalised on the basis of the active root depth (Laio et al. 2001a):

[eta] = [E.sub.max]/Po[Z.sub.r] (8)

[[eta].sub.w] = [E.sub.max]/Po[Z.sub.r] (9)

m = [K.sub.s]/Po[Z.sub.r][[[e.sup.[beta](1-[s.sub.e_fc])] - 1] (10)

1/[gamma] = [alpha]/Po[Z.sub.r] (11)

[lambda]' = [lambda] [e.sup.-[DELTA]/[alpha]] (12)

An expression for the constant c can be obtained by integration in order to obtain an area under the p([s.sub.e]) function that has a value of 1 (Laio et al. 2001a).

Parameter determination

Additional soil samples were collected in order to determine the parameters in the stochastic model from measured data. Nine undisturbed soil samples were taken from a depth of 0.3-0.4 m in each plot using stainless steel cutting rings that were 0.05 m long and 0.05 m in diameter. These samples were used to: (1) produce soil water retention curves using the centrifugation method (Klute 1986) (Hitachi CR21G centrifuge; 20[degrees]C) at suctions of 0.001, 0.005, 0.01, 0.02, 0.04, 0.06, 0.08, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0 and 1.5 MPa, from which field capacity and other important soil water parameters could be determined. The empirical equations between [S.sub.e] and soil suction ([psi]) for three vegetation types are as follows:

[mathematical expression not reproducible] (13)

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

(2) calculate soil bulk density based on the volume of the soil core and its mass after oven-drying at 105[degrees]C to constant mass (ASTM C29/C29M-09 2003); and (3) determine [K.sub.s] using the constant head method (Klute and Dirksen 1986; Liu et al. 2007). For each soil core, the water retention curves and [K.sub.s] were determined for three replications and the averaged values were used. The Po was calculated from the bulk density and an assumed particle density of 2.65 g [cm.sup.-3] (Wang et al. 2013). The values of [s.sub.e_h], [s.sub.e_w], [s.sub.e.sup.*] and [s.sub.e_fc] for each vegetation type were calculated using Eqns 13 and 14 on the basis of the following soil water potentials: [[psi],se_h] = -10 MPa; [[psi],se_w] = -3 MPa; [mathematical expression not reproducible] = -0.03 MPa; and [[psi],se_fc] = -0.01 MPa (Hillel 1998) respectively. The value of [beta] was determined from the water retention curve according to Laio et al. (2001a).

For each plot, [Z.sub.r], defined as the soil depth range in which 95% of the below-ground biomass was distributed, was determined by field measurements (Laio et al. 2001a; Liu et al. 2007). Leaf area index (LAI) was measured by a plant canopy analyser (LAI-2000, LI-COR, USA) two times each month during the growing season, and the LAI value used to determine the canopy interception (cm), [DELTA] = 0.043 x LAI. (Clapp and Hornberger 1978; Laio et al. 2001a; Whitehead and Beadle 2004; Vervoort and van der Zee 2009).

The daily rainfall amount was measured at the weather station, and the frequency of rainfall was analysed on daily time scale. The mean daily evapotranspiration rate under unrestricted soil water conditions ([E.sub.max]) was calculated from the daily meteorological data using the Penman-Monteith equation (Caylor et al. 2005; Franz et al. 2010). The mean daily evapotranspiration rate when [s.sub.e] = [s.sub.e_w] ([E.sub.w]) was determined by using an empirical equation: [E.sub.w] = 0.05[E.sub.max] (Franz et al. 2010). All model parameter values are shown in Table 2.

Statistical analyses

The [S.sub.e] was discretised by the resolution of 0.01, and the number and frequency of values in each moisture interval were counted and calculated. Descriptive statistical properties--i.e. minimum, maximum, mean, median, standard deviation (s.d.) and coefficient of variation (CV)--were determined in order to explore the basic characteristics of the relative soil water content distributions. The peak value was derived from the point of [s.sub.e] at which was the highest p(se) value. Two statistical criteria were used to evaluate the performance of the modified Laio model: the mean deviation (MD) and the root mean square error (RMSE).

MD = 1/n [N.summation over (i=1)] [absolute value of [P.sub.i] - [O.sub.i]] (16)

RMSE = [square root of 1/n [N.summation over (i=1)] [([O.sub.i] - [P.sub.i]).sup.2]] (17)

where [P.sub.i] is the zth predicted probability value of p([s.sub.e]), [O.sub.i] is the ith observational probability value of p([s.sub.e]), n is the number of data pairs and [s.sub.e] ranges within 0.01-1 corresponding to n of 1-100.

A correlation analysis was also used to compare the measured and simulated values of p([s.sub.e]). Analysis of variance with a least significant difference test at the 95% confidence level was used to compare the mean values of relative soil water content for different vegetation types. All statistical determinations were made using SPSS 17.0 software (SPSS Inc. 2008). The Laio model was run using MATLAB 2018 software (MathWorks 2018).


Descriptive statistics of relative soil water contents

The descriptive statistics of the mean relative soil water contents of the 0.8, 4.0 and 4.0 m profiles measured under the three vegetation types (BOI, SEB and CHP) during the study period are presented in Table 3. Differences in the mean relative soil water contents under the three vegetation types were all significant (P < 0.05), and the differences between these values were all greater than 3%. In the three vegetation plots, mean relative soil water contents under BOI had the highest mean and median values (0.39 and 0.40 respectively), those under CHP had intermediate values (0.36 and 0.36) and those under SEB were the lowest (0.32 and 0.33).

The maximum relative soil water content was considerably higher under BOI (0.6) than under the other two vegetation types; the values under CHP and SEB were similar (Table 3). The main reason was that BOI had an active root depth of 0.8 m while CHP and SEB had an active root depth of 4 m. The minimum relative soil water content was similar under all three vegetation types (0.21-0.24). The degree of variation in relative soil water content, as indicated by both s.d. and CV, was ranked in order: BOI > CHP > SEB.

Effects of rainfall and PET variations on soil water dynamics

Daily variations in rainfall and measured relative soil water content under the three different vegetation types are shown in Fig. 2. At the beginning of the growing season (April-June), relative soil water contents decreased due to the effects of vegetation growth and the limited rainfall to the lowest values during June-August. After this time, relative soil water contents tended to increase with the increasing rainfall amount, and were often stable towards the end of the growing season. The degree to which relative soil water contents decreased or increased varied among the years, with those in 2011 exhibiting the highest rates of decrease and increase.

The measured surface runoff varied, ranging within 1.4-2.0 with a mean of 1.8 mm x [year.sup.-1] for BOI, 5.4-6.7 with mean of 5.9 mm/year for CHP and 2.5-2.9 with mean of 2.7 mm x [year.sup.-1] for SEB. Because surface runoff was less than 2% of the total amount of rainfall during the growing season, its effect on soil water dynamics could be ignored.

The daily values of PET and the relative soil water content under the three different vegetation types are shown in Fig. 3. From April to June in 2011 or July in 2013 and 2014, the PET increased steadily with the warmer temperatures to a maximum exceeding 8 mm x [d.sup.-1], and then decreased to a minimum in November, which was at the end of the growing season.

The PET tended to increase from the beginning of the growing season in April and then decreased until the end of the growing season in November (Fig. 3). The changes in PET did not exactly correspond to those in rainfall amount. For example, maximum PET occurred in June or July, but in 2011 and 2014 the maximum rainfall was in September. The changes in daily PET were less than those in daily rainfall amount and differed little among the three study years, but had a positive correlation with LAI changes which increased from the beginning of the growing season to August and decreased from August to the end of the growing season. Thus, PET was affected more by the steady changes in vegetation growth than by the rainfall or climate in the three study years. The daily rainfall exhibited a greater degree of fluctuation and also varied among the three study years, which resulted in irregular trends in soil water in those years, indicating that it affected soil water to a greater extent than PET.

Stochastic simulation of soil water dynamics

All the parameters in Table 2 were used as the inputs for the Laio model (Eqn 6) after standardisation (Eqns 7-11), and the PDF for each vegetation type was simulated. Figure 4 presents a comparison of the simulated and measured PDFs for each vegetation type. The stochastic model described the soil water dynamics reasonably well for the three vegetation types during the three growing seasons at the point scale in the study area. In particular, the model well simulated the peak position and the range of soil water PDF values, and the simulation was better as the rooting depth of the vegetation type increased. However, there were some inconsistencies in the measured relative soil water content PDFs, as depicted in the observed histogram frequency distributions, and those that were predicted by the model simulations. For example, when the relative soil water content was less than 0.3, the p([s.sub.e]) values that the model simulated were always less than the observed values for all three vegetation types.

In order to verify the accuracy of the simulated PDF for the three vegetation types, the statistical moments of the relative soil water contents of the simulated and measured PDF were compared (Table 4). The maximum values of the simulated PDF were approximately the same as the corresponding measured values for all vegetation types, with the largest difference between these values being 0.02. However, accuracy of the minimum values produced by the simulation were slightly lower, with differences between the simulated and the smaller measured values of 0.03, 0.05 and 0.06 for BOI, SEB and CHP respectively. The peak value of the simulated PDF was significantly lower than the measured value for BOI, but the simulated peak and measured values were the same for SEB and CHP (Fig. 4). The MD values followed the order BOI > SEB > CHP (0.464, 0.334 and 0.269 respectively), and the order of RMSE values was SEB > BOI > CHP (1.051,0.973 and 0.832 respectively). As noted above, the differences between the measured and simulated results might be due to some simple assumptions in the stochastic model when predicting lower relative soil water contents.

Sensitivity of relative soil water content PDF to vegetation type

Given that the climate, topography and initial soil characteristics were the same for each plot, the PDF of the relative soil water content simulated by the stochastic model for each plot was only affected by vegetation type. The stochastic model had 13 parameters, and six of these were related to the vegetation: [Z.sub.r] (active root depth), [DELTA] (canopy interception), [s.sub.e_w] (permanent wilting point), [s.sub.e.sup.*] (relative soil water content below which plants begin closing their stomata), [E.sub.max] (mean daily evapotranspiration rate at [s.sub.e_fc]) and [E.sub.w] (mean daily evaporation rate at [s.sub.e_w]). The following procedure was used to test the sensitivity of the model prediction to the different parameters. Taking the BOI plot as an example, the changes in PDF were simulated while one of the parameter values was either increased or decreased by 10%, and the values of the remaining five parameters remained unchanged. This procedure was repeated for each of the six parameters in turn. The simulated results are shown in Fig. 5.

Among the six parameters related to vegetation type, the PDF of the relative soil water content was not sensitive to changes in values of either [DELTA] or [E.sub.w] (Fig. 5). This is clear because the shape of the PDF presented no noticeable changes when the values of [DELTA] and [E.sub.w] were either increased or decreased by 10% from their base values for BOI. In contrast, the PDF of the relative soil water content was sensitive to changes in the values of either [s.sub.e.sup.*] or [E.sub.max]; the PDF moved to the right and the peak was reduced with increases in the value of [s.sub.e.sup.*], but it moved to the left and the peak increased with increases in [E.sub.max]. The PDFs of the relative soil water content were moderately sensitive to changes in the values of [Z.sub.r] and [s.sub.e_w], compared with the other two pairs of parameters. Increasing the [Z.sub.r] value resulted in a slight narrowing of the base of the PDF, and the peak noticeably increased. Increasing [s.sub.e_w] moved the PDF slightly to the right and increased the peak.

The changes in the pattern caused by changing the value of one of the four parameters ([Z.sub.r], [s.sub.e_w], [s.sub.e.sup.*] and [E.sub.max]) was consistent with the differences in the soil water PDFs under the three different vegetation types. Large differences in the values of [Z.sub.r], which were obviously related to the three vegetation types was the main cause of the observed changes in the peaks and widths of the PDFs (Fig. 4). Therefore, the peak of the PDF was lower for BOI than for SEB and CHP, but the PDF distribution was wider for BOI than for SEB and CHP. The values of [s.sub.e_w], [s.sub.e.sup.*] and [E.sub.max] mainly affected the relative positions of the PDFs along the x-axis (related to the relative soil water content). The combination of [Z.sub.r], [s.sub.e.sup.*] and [E.sub.max] for the BOI plot were mainly responsible for the distinct differences in the shape of the PDF compared with the SEB and CHP plots.

Determination of sustainable vegetation types

The Laio model was used to determine the most sustainable vegetation types by considering the soil water availability among the three plant species. Figure 6 shows the simulated soil water PDFs for three vegetation types under four LAI scenarios. With the same LAI, the BOI always had the highest median of soil water PDF, followed by CHP and then SEB. For the same vegetation type, the median of soil water PDF decreased with increasing LAI.

The stable water content is defined as a threshold at which soil water movement towards the evaporation surface ceases due to limitations imposed by low hydraulic conductivity. In the Loess Plateau, the stable water content is generally considered to be a criterion to assess soil desiccation, which is usually calculated using 65% of field water-holding capacity and its [s.sub.e] value equal to 0.325 in the study area (Chen et al. 2010; Wang et al. 2010). The probability of [s.sub.e] < 0.325 was 2.4, 8.1, 14.2 and 20.1% for BOI for LAI values of 1.0, 1.5, 2.0 and 2.5 respectively; and, correspondingly for the same LAI values, 28.8, 62.0, 78.9 and 88.2% for SEB and 0.7, 7.0, 18.4 and 31.7% for CHP (Fig. 6). Among the three vegetation types, SEB had the largest probability with [s.sub.e] < 0.325 and most easily caused soil desiccation. Therefore, SEB was less suitable to grow in the study area than BOI and CHP.


Statistics of relative soil water contents

In this study, the magnitude of the relative soil water contents under the three vegetation types were BOI > CHP > SEB (Table 3), similar to previous findings for these vegetation types on the Loess Plateau (Fu et al. 2006; Chen et al. 2010; Wang et al. 2010, 2012a). The differences were related to the comprehensive influences of the mean LAI during the growing season, the root water uptake rate, active root system distribution and the depth of rainwater infiltration. The large LAI usually reflects greater water consumption by plants and consequently results in low relative soil water contents in the active root zone. During the observed periods, mean LAI was 1.36 for BOI, 1.50 for SEB and 1.64 for CHP (Table 2). Under water stress conditions, the root water uptake rate was limited by the potential transpiration rate (a root water uptake stress response function), relative soil water content and the root distribution in the soil profile (Huang et al. 2015). A discrete function based on the two pressure heads at which the plant begins to suffer water stress and ceases to take up water respectively, has often been used to describe the response of root water uptake to water stress (Feddes et al. 1974). Low pressure heads mean that the response of root water uptake to water stress is slow and that the plant can take up more water from soil. Based on the results of Wesseling and Brandyk (1985), Kimball et al. (1997), Dang et al. (1997), Xia and Shao (2008) and Zhang et al. (2018), the pressure head at which the plant began to suffer from water stress was -150 kPa for BOI and -500 kPa for SEB and CHP; and the pressure head at which plants stopped water uptake was -800 kPa for BOI, -1500 kPa for SEB and -1700 kPa for CHP. The active root depth was 80, 400 and 400 cm for BOI, SEB and CHP (Table 2), which would result in potentially greater depth proportions of rainwater recharge for the same rainfall events under BOI than under SEB and CHP (Wang et al. 2012b). Therefore, under the same soil, precipitation and PET conditions, the SEB plot always had the lowest relative soil water content followed by CHP, and BOI had the highest.

Effects of rainfall and PET variations on soil water dynamics

In the study area, rainfall amount and its daily variation was the main water input and, thus, is one of the most important factors affecting soil water dynamics. The precipitation events on the Loess Plateau can be divided simply into two categories: small rainfall events occurring at a higher frequency and large rainfall events at a lower frequency. The total of small rainfall events showed little variation during the rainy season, while the total of large rainfall events exhibited markedly greater variation, which resulted in the large inter-annual and annual variations observed in the amount of precipitation in the study area (Chen et al. 2008; Wang et al. 2012a). Small rainfall events affect only the soil water dynamics in shallow soil layers (0-40 cm), but large events are the main contributors affecting soil water dynamics in deeper layers (40-500 cm) (Xie et al. 2015). Evapotranspiration represents a large component in the water balance equation in the study area, and includes evaporation from the soil and transpiration from vegetation (Li et al. 2016b). A large fraction of the soi 1 water in the surface layer is lost through direct soil evaporation, while the rates of plant water uptake increase and soil evaporation rates decrease in the deeper soil layers (Wang et al. 2012a). Our observation periods were focused on the rainy seasons when relative soil water contents were higher. During these periods, BOI had lower LAI of 1.36 and could not protect the soil surface from direct solar radiation, leading to greater daily water losses via direct evaporation, whereas CHP had a higher LAI of 1.64 that reduced direct evaporation but resulted in soil water losses mainly via transpiration.

Stochastic simulation of soil water dynamics

Previous studies demonstrated three different methods for the simulation of soil moisture dynamics: mathematical statistical, soil hydrodynamic and stochastic model methods (Sheikh et al. 2009). The mathematical statistical method, which is based on empirical relationships between environmental variables (e.g. soil properties, topography and vegetation characteristics) and soil moisture content, is relatively simple in structure and lacks a physical explanation (Suo et al. 2018). The physically-based soil hydrodynamic method, such as HYDRUS (Simunek et al. 2012) and WAVES (Zhang et al. 1996) models, requires several soil parameters to be optimised for accurate simulation of soil moisture dynamics, which can be experimentally expensive and time consuming. Finally, the stochastic model takes soil moisture as a random variable. This is affected by several factors (e.g. rainfall, evapotranspiration and leakage). In this method, the PDF of soil moisture is constructed by combining the soil moisture balance equation with random analyses of rainfall, evapotranspiration, leakage and other processes (Laio et al. 2001a). Compared with the other two methods, the stochastic model can characterise soil moisture distributions and provide range, median and frequency of soil moisture distribution for a specific soil texture and vegetation type.

In this study, the Laio stochastic model described reasonably well the PDF of soil water dynamics for the three vegetation types at the point scale, especially under CHP with its deep root system distribution, although some discrepancies between the simulated and observed PDFs remained for all three vegetation types. The stochastic description of soil water dynamics has often been thought of as a minimalistic description of the soil water balance, through which the probabilistic characteristics of soil water dynamics can be derived analytically by inputting the statistical information of precipitation, PET and other parameters. Generally, a stochastic model is developed as an efficient tool for analysis of ecohydrological implications based on statistical soil water data (Rodriguez-Iturbe and Porporato 2005) rather than as a means of prediction. However, a predictive stochastic model can be developed if the modelling framework is improved (Laio et al. 2001a, 20016), including: (1) daily infiltration and redistribution processes of soil water; (2) soil water drainage process when soil water exceeds field capacity; and (3) the vertical distributions of soil water and root system. In reality, soil water varies in a soil profile due to the interactive effects of rainfall infiltration, soil evaporation, root water uptake and soil water redistribution. Although the model used in this study was largely based on the assumptions given above and using statistically-averaged growing seasons, the good agreement between the soil water PDFs derived from the results of the model and the field observations showed that the model can be reasonably applied to predict soil water dynamics in this region.

Sensitivity of relative soil water content PDF to vegetation type

We concluded that the Laio stochastic model accurately simulated the soil water PDFs under all three vegetation types in the semiarid study area on the Loess Plateau. However, soil water dynamics on the Loess Plateau are very complex due to the various types of vegetation cover, climatic conditions and soil texture characteristics. Future research should consider the effects of climate and soil texture characteristics on soil water dynamics, which can be simulated by changing the parameters in the Laio model.

Rainfall and evapotranspiration affect soil water dynamics (Huang and Gallichand 2006; Chen et al. 2008; Wang et al. 2012b). Their impacts can be reflected by changes in the parameters in the Laio model. The values of a and [lambda] increase with increases in rainfall amount and the values of [E.sub.max] and [E.sub.w] increase with increases in evapotranspiration. The resulting PDF of the relative soil water contents moves to the right and the peak value is reduced with the increases in a and [lambda] or with the decreases in the [E.sub.max] (Laio et al. 2001 a, 20016).

The soil texture has an effect on the values of the [beta], Po, Zn [K.sub.s], [s.sub.e_h], [s.sub.e_w], [s.sub.e.sup.*], [s.sub.e_fc], [E.sub.max] and [E.sub.w] parameters in the Laio model, but the main impact is embodied in the soil waterholding performance that is closely related to the values of parameters [s.sub.e_h], [s.sub.e_w], [s.sub.e.sup.*] and [s.sub.e_fc] (Laio et al. 2001a). It is obvious that lower relative soil water contents are much more likely to occur in sandy than in silty loam soils and clay soils. The PDF shifts to higher relative soil water content values with increasing clay content of soil (Laio et al. 2001 b).

Determination of sustainable vegetation types

Simulation results indicated that the most sustainable vegetation types were BOI and CHP compared with SEB by considering soil water availability. This was mainly due to the high water consumption of SEB in the deep layer. Chen et al. (2010) found that the relative soil water content under SEB was significantly lower than CHP and semi-natural grassland at depths below 1 m, and Yao et al. (2012) found that soil water content under SEB was lower than under grassland and Robinia pseudoacacia woodland on the Loess Plateau. Thus SEB should be carefully considered before use for reforestation on the Loess Plateau. Increasing plant coverage can significantly reduce soil loss, but high plant coverage easily results in soil desiccation due to high water consumption. Optimal plant coverage is vital to balance soil water consumption and control soil loss. Snelder and Bryan (1995) found that soil loss was a maximum for plant coverage of below 25%, and that a minimum of 55% coverage was required to satisfactorily control soil erosion in woodland. Garcia-Ruiz et al. (1995) and Molinillo et al. (1997) showed that soil loss increased rapidly when reducing plant coverage, and that little soil loss occurred for shrub coverage exceeding 50%. Therefore, the optimal plant coverage (given by mean maximum LAI) should be greater than 1.3 for BOI, SEB and CHP based on the relationship between plant coverage (M) and LAI (M = 1 - [e.sup.-0.62 LAI]) (Huang et al. 2011). For LAI = 1.3, the cumulative probability of relative soil water contents less than the stable value was 6% for BOI, 50% for SEB and 5% for CHP (Fig. 6). Compared with BOI and CHP, SEB was easily subject to water stress and resulted in soil desiccation and was not a sustainable vegetation type for the study area.

Future work should use soil water measurements made in soils with different textures and in different climatic regions on the Loess Plateau to assess the performances of the Laio model and use the stochastic model to study the effects of different textures and climatic on vegetation types (Rodriguez-Iturbe and Porporato 2005). The stochastic model could be a valuable tool in developing optimal vegetation restoration strategies for sustainable plant growth in various climate and soil conditions on the Loess Plateau.


The probabilistic distribution of soil water dynamics at the point scale was studied under three dominant vegetation types on the Loess Plateau using the stochastic model improved by Laio et al. (2001 a). The main results and conclusions follow:

(1) The mean relative soil water content in the study area differed in the following order: BOI > SEB > CHP. Soil water was clearly related to the amounts of rainfall and evapotranspiration on a daily scale, respectively the most important input and output terms, of the stochastic model.

(2) The shape of the PDF curves and the distribution characteristics of the relative soil water contents showed good agreement between simulated and observed p(se) values.

(3) Under the same climate and topography conditions of the study area, the simulated soil water PDF was most sensitive to changes in [s.sub.e.sup.*] and [E.sub.max], was moderately sensitive to changes in [Z.sub.r] and sew and was not sensitive to and [E.sub.w]. The [Z.sub.r] value mainly affected the peak and width of the soil water PDF, while the [s.sub.e_w], [s.sub.e.sup.*] and [E.sub.max] mainly affected the values of relative soil water content. The [s.sub.e_w] and [s.sub.e.sup.*] values were positively correlated, and negatively correlated, with the relative soil water content. Differences in the values of [Z.sub.r], [s.sub.e_w], [s.sub.e.sup.*] and [E.sub.max] accounted for the differences in the shapes of the relative soil water content PDFs under the three vegetation types.

(4) The Laio model could be applied to the soil water dynamic simulation in the semi-humid study area of the Loess Plateau and was especially accurate for soil water dynamic simulations under vegetation with deeper roots. Based on the stochastic distributions of soil water dynamics for different LAI values, the BOI and CHP were the most sustainable vegetation types compared with SEB by considering soil water availability and soil loss control. Future work is necessary to assess the performance of the Laio model under different soil and climatic conditions.

Conflicts of interest

The authors declare no conflicts of interest.


Financial assistance for this work was provided by the National Natural Science Foundation of China (No. 41571130082,41390463 and 41571213). Special thanks are given to Prof. Dr Mick Whelan, associate editor and two anonymous reviewers for their comments and suggestions.


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Handling editor: Mick Whelan

Li zhu Suo (A) and Mingbin Huang (ID) (A,B)

(A) State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau, Institute of Soil and Water Conservation, Northwest A & F University, Yangling, 712100, China.

(B) Corresponding author. Email:

Caption: Fig. 1. Behaviour of soil water losses (evapotranspiration and leakage), as a function of relative soil water contents for typical climate, soil and vegetation characteristics in semiarid ecosystems (Laio et al. 2001a).

Caption: Fig. 2. Daily total rainfall (bars) and relative soil water content (lines) during the growing seasons in the three study years.

Caption: Fig. 3. Daily values of potential evapotranspiration (PET) calculated using the Penman-Monteith equation (black line) and relative soil water content (coloured lines) during the growing seasons in the three study years.

Caption: Fig. 4. Comparison of simulated and measured soil water probability density functions under three types of vegetation. (BOI, Bothriochloa ischaemum L.; SEB, sea-buckthorn; CHP, Chinese pine); p(se) is the probability density.

Caption: Fig. 5. Changes in the soil water probability density function of BOI with parameter variation: A (canopy interception, cm), [E.sub.w] (mean daily evaporation rate at [s.sub.e_w], cm x [d.sup.-1]), [s.sub.e.sup.*] (relative soil water content below which plants begin closing their stomata. dimensionless), [E.sub.max] (mean daily evapotranspiration rate at [s.sub.e_fc], cm x [d.sup.-1]), [Z.sub.r] (active root depth, cm) and [s.sub.e_w], (permanent wilting point, dimensionless); p([s.sub.e]) is the probability density.

Caption: Fig. 6. Changes in the soil water probability density function under three types of vegetation with LAI variation. (BOI, Bothriochloa ischaemum L.; SEB, sea-buckthorn; CHP, Chinese pine); dotted line is the stable relative water content of 0.325; p([s.sub.e]) is the probability density.
Table 1. Summary of relevant studies that used the
Laio and/or modified Laio stochastic models of
soil water dynamics

Study                     Active root      Vegetation type
                          depth (cm)

Laio et al. (2001b)           100          Tree and grass

Porporato et al. (2003)     40-100         Tree and grass
Liu et al. (2007)             40                Grass
Pumo et al. (2008)            150               Tree
Teuling et al. (2009)         60                Tree
Tamea et al. (2010)           20            Wetland shrub
Huang et al. (2013)           60                Shrub
Li et al. (2016a)           5-22.5       Bare soil and shrub
This study                  80, 400     Grass, shrub and tree

Study                          Climate               Soil type

Laio et al. (2001b)          Semi-humid                Sand
Porporato et al. (2003)      Arid-humid                Sand
Liu et al. (2007)               Arid             Montace grey soil
Pumo et al. (2008)              Humid         Loamy sand, sandy loam,
Teuling et al. (2009)      Arid, humid and    clay Sand, loam, clay,
                          transitional zone     coarse, peat, etc.
Tamea et al. (2010)              --              Loam, marly-peat
Huang et al. (2013)             Arid                   Sand
Li et al. (2016a)               Arid           Sand and gravel plain
This study                    Semiarid            Silty clay loam

Study                                   Location

Laio et al. (2001b)                Savanna of Nysvley,
                                      South Africa
Porporato et al. (2003)          Kalahari, South Africa
Liu et al. (2007)                Qilian Mountain, China
Pumo et al. (2008)            Eleuterio River basin, Italy
Teuling et al. (2009)                      --
Tamea et al. (2010)           Everglades National Park, USA
Huang et al. (2013)       Shapotou Experimental Station, China
Li et al. (2016a)                 Namib Desert, Namibia
This study                        Loess Plateau, China

Table 2. Values of the stochastic model
parameters for three vegetation types
BOI, Bothriochloa ischaemum L.; SEB, Sea-buckthorn;
CHP, Chinese pine. All water contents are expressed
as the relative soil water content, [s.sub.e].

Parameters                      Symbols               Units

Empirically fitted              [beta]                  --
parameter of soil water
retention curve

Soil porosity                     Po                    --

Saturated hydraulic            [K.sub.s]         cm x [d.sup.-1]

Active root depth              [Z.sub.r]                cm

Aboveground vegetation             h                    m

Maximum leaf area index           LAI         [m.sup.2] x [m.sup.-2]

Canopy interception            [lambda]                 cm

Maximum stomatal             g[s.sub.max]       mmol x [m.sup.-2]
conductance (D)                                     [s.sup.-1]

Hygroscopic point             [s.sub.e_h]               --

Relative soil water at        [s.sub.e_2]               --
permanent wilting point

Relative soil water         [s.sub.e.sup.*]             --
content at which plants
begin closing stomata

Relative soil field          [s.sub.e_fc]               --

Mean daily                    [E.sub.max]        cm x [d.sup.-1]
evapotranspiration rate
when [s.sub.e] =

Mean daily                     [E.sub.w]         cm x [d.sup.-1]
evapotranspiration rate
when [s.sub.e] =

Mean rainfall depth             [alpha]                 cm

Arrival rate of rainfall       [lambda]             [d.sup.-1]

                                     Parameter values

Parameters                   BOI       SEB       CHP

Empirically fitted           15.0     15.0      15.0
parameter of soil water
retention curve

Soil porosity                0.51     0.51      0.51

Saturated hydraulic          91.0     91.0      91.0

Active root depth           80 (A)   400 (B)   400 (C)

Aboveground vegetation       0.4       2.0       4.0

Maximum leaf area index      1.36      1.5      1.64

Canopy interception         0.058     0.065     0.070

Maximum stomatal             400       180       150
conductance (D)

Hygroscopic point            0.15     0.15      0.15

Relative soil water at       0.24     0.20      0.22
permanent wilting point

Relative soil water          0.56     0.44      0.52
content at which plants
begin closing stomata

Relative soil field          0.65     0.65      0.65

Mean daily                  0.417     0.437     0.432
evapotranspiration rate
when [s.sub.e] =

Mean daily                  0.021     0.022     0.022
evapotranspiration rate
when [s.sub.e] =

Mean rainfall depth          0.75     0.75      0.75

Arrival rate of rainfall     0.32     0.32      0.32

(A) Cheng et al. (2009).

(B) Ruan and Li (2002).

(C) Duan et al. (2016a).

(D) Caylor et al. (2005).

Table 3. Descriptive statistics of relative soil
water contents under three vegetation types
BOI, Bothriochloa ischaemum L.; SEB, Sea-buckthorn; CHP, Chinese
pine, s.d., standard deviation; CV, coefficient of variation.
Mean values followed by the same letter are not significantly
different among vegetation types (P < 0.05, least
significant difference test).

                         Relative soil water content
Vegetation   Sample
type         number   Minimum   Maximum   Mean    Median

BOI           159      0.24      0.60     0.39a    0.40
SEB           159      0.21      0.40     0.32c    0.33
CHP           159      0.24      0.43     0.36b    0.36

Vegetation   s.d.     CV
type                 (%)

BOI          0.069   0.18
SEB          0.040   0.13
CHP          0.036   0.10

Table 4. Eigenvalue analyses between simulated and measured values
of the relative soil water content and soil water probability
density (PDF) under three vegetation types
BOI, Bothriochloa ischaemum L.; SEB, Sea-buckthom;
CHP, Chinese pine. MD, mean deviation; RMSE, root mean square error.
The standard used for the simulated maximum and minimum values is
that the probabilities are greater than 0.05

                                  Eigenvalues of relative
Vegetation    Source of           soil water content
type         eigenvalues
                           Minimum   Maximum   Peak   Range

BOI           Measured      0.24       0.6     0.42   0.37
              Simulated     0.27      0.62     0.38   0.36
SEB           Measured      0.21      0.40     0.33   0.23
              Simulated     0.26      0.41     0.32   0.16
CHP           Measured      0.24      0.43     0.36   0.20
              Simulated     0.30      0.45     0.36   0.16

Vegetation    Source of     PDF
type         eigenvalues
                            MD     RMSE

BOI           Measured     0.464   0.973
SEB           Measured     0.334   1.051
CHP           Measured     0.269   0.832
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Author:Suo, Li zhu; Huang, Mingbin
Publication:Soil Research
Article Type:Report
Geographic Code:9CHIN
Date:Aug 1, 2019
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