The stochasticity of soil erosion reflects the variability of soil hydrological
response to precipitation in a complex environment. Assessing this
stochasticity is important for the conservation of soil and water resources; however, the
stochasticity of erosion event in restoration vegetation types in
water-limited environment has been little investigated. In this study, we
constructed an event-driven framework to quantify the stochasticity of
runoff and sediment generation in three typical restoration vegetation types
(

Soil erosion is a global environmental problem. In recent centuries, the erosion rate worldwide has been accelerating due to climate change and anthropogenic activities, causing soil deterioration and terrestrial ecosystem degradation (Jiao et al., 1999; Marques et al., 2008; Fu et al., 2011; Portenga and Bierman, 2011). The uncertainty and intensity of soil erosion are major features of the erosion phenomenon. Although many studies have concentrated on the intensity of erosion at different spatiotemporal scales (Cantón et al., 2011; Puigdefábregas et al., 1999), the uncertainty of soil erosion generation is a further challenge for researchers working to improve the accuracy of erosion prediction. The stochasticity of environment and spatiotemporal heterogeneity of soil loss is the main influence on the randomness of runoff production and sediment transportation in natural conditions (Kim et al., 2016). But the complex mechanism of erosion generation also increases the uncertainty and variation of erosion processes (Sidorchuk, 2005, 2009). Therefore, how to effectively describe erosion stochasticity and to reasonably assess its impacting factors is necessary and important for understating soil erosion science from the perspective of randomness.

First, combinations of various probabilistic, conceptual and physical models have been reported as different integrated approaches to describe the stochasticity of soil erosion intensity (see Table 1). As one form of erosion intensity, the runoff has been shown as a stochastic process by different mathematic simulation models. Some studies (Moore, 2007; Janzen and McDonnell, 2015) have also simulated the stochasticity, and further quantified the randomness of runoff production and its connectivity dynamics in hillslope and catchment scales by using different probabilistic distribution functions and conceptual models. In these studies, the theory-driven conceptual models simplified the main hydrological behaviours related to runoff production, highlighting the stochastic effects of infiltration and precipitation on runoff processes. Based on the above precondition, the data-driven probabilistic models further simulated the stochastic runoff production by mapping or calibrating the difference between observed and predicted probabilistic values. As a result, the stochastic-conceptual approaches have formed an effective framework to model rainfall–runoff processes (Freeze, 1980), as well as to assess flood forecasting (Yazdi et al., 2014).

Summary of the research on the stochasticity of soil erosion rate and the stochasticity of factors affecting soil erosion rate.

The stochasticity of soil erosion rate which is another pattern of erosion intensity has been investigated by probabilistic and physical models in some studies. The theory-driven physical models in these studies (Sidorchuk, 2005) integrated hydrological and mechanical mechanisms of overflow and soil structure with sediment transpiration processes, stressing the stochastic effect of physical principles on erosion rate in different spatial scales (Table 1). Sidorchuk (2005) introduced stochastic variables and parameters into probabilistic models by randomizing the physical properties of overflow and soil structure. This approach developed the understanding of uncertainty of sediment transpiration processes, causing the randomness simulation to better fit the reality of stochastic erosion rate (Sidorchuk, 2009). Additionally, the stochasticity of soil erosion rate also reflected the erosion risk which was assessed by the integration of a theory-driven empirical model with geostatistics (Jiang et al., 2012; Wang et al., 2002; Kim et al., 2016). Erosion risk analysis has generally concentrated on the uncertainty or variability of soil erosion rate at catchment and regional scales, highlighting the impact of the spatiotemporal heterogeneous rainfall and other environment conditions on the stochastic erosion rate. In summary, these probabilistic and physical models constituted a systematical analysis framework closely related to the principle of water balance and basic hydrological assumptions. This effectively described the randomness of soil erosion rate affected by complex hydrological processes (Bhunya et al., 2007). However, few studies have been made to analyse the stochasticity of soil erosion events. In particular, there has been little effort to systematically investigate how the signal of stochastic rainfall is transmitted to erosion events occurring in different restoration vegetation types based on observational data rather than on other model assumptions. Yet such event-based investigation deriving from specific experiment results may be more practically meaningful for understanding the stochastic interaction between rainfall and erosion events.

Secondly, the probabilistic approaches have also been reported as a crucial tool to describe the stochasticity of factors affecting soil erosion rate (Table 1). The randomness of soil water content (Ridolfi et al., 2003), antecedent soil moisture (Castillo et al., 2003), infiltration rate (Wang and Tartakovsky, 2011) and soil erodibility (Wang et al., 2001) in heterogeneous soil types have all been modelled by different probability distribution functions. The stochasticity of soil hydrological characteristics related to erosion rate mainly impacted in various ways the spatiotemporal distribution of erosion rate, especially at regional or larger spatial scales. Meanwhile, as the main driving force of soil erosion generation, the uncertainty of rainfall event to some extent represents the environment stochasticity (Andrés-Doménech et al., 2010). Eagleson in 1978 applied probabilistic-trait models to characterise the stochasticity of rainfall event by using Poisson and Gamma probability distribution functions. The stochastic rainfall distribution in different spatiotemporal scales has also been applied to examine the effect of runoff and sediment yield (Lopes, 1996), to calibrate the runoff–flood hydrological model (Haberlandt and Radtke, 2014), as well as to evaluate sewer overflow in urban catchment (Andrés-Doménech et al., 2010).

The role of spatial distribution of vegetation in controlling the soil erosion rate under different spatiotemporal scales has been well recognized (Wischmeier and Smith, 1978; Puigdefábregas, 2005). How the plants reduce soil erosion rate has also been illuminated and interpreted by various physical and empirical models (Liu, 2001; Mallick et al., 2014; Prasannakumar et al., 2011). In theory, Puigdefábregas (2005) proposed vegetation-driven spatial heterogeneity (VDSH) to explain the relationship between vegetation patterns and erosion fluxes, which improves understanding of the hydrological function of plants in erosion processes. The trigger–transfer–reserve–pulse (TTRP) framework proposed by Ludwig in 2005 systematically explored the responses and feedback between vegetation patches and runoff erosion during ecohydrological processes. Theoretically, the stochastic signals of different rainfall events could also be disturbed by the hydrological function of plants, finally affecting the randomness of runoff and sediment events occurring in various vegetation types. However, little effort has been made to explore the effect of different vegetation types on the stochasticity of soil erosion events. In particular, few approaches have been used to analyse how the properties of rainfall, soil and vegetation impact on the stochastic erosion events through event-based investigation deriving from observational data rather than via theory-based models. Actually, logistic regression modelling (LRM) probably deals with the above problems. LRM evaluates the causal effects of categorical variables on dependent variables, and quantifies the probabilistic contribution of influencing factors on the randomness of responsive random events in terms of an odds ratio (Hosmer et al., 2013). This can be seen as another probabilistic model to explore the probability attribution of influencing factors. However, little literature is available on LRM being used to explore the probabilistic attribution of stochastic erosion events under complex environmental conditions.

In this study, we have hypothesized that the uncertainty of erosive events was also an important property of the soil erosion phenomenon, and monitored erosion events occurring in three typical restoration vegetation types at runoff plot scale over five consecutive years' rainy seasons. We aim to (1) comprehensively describe the stochasticity of runoff and sediment events in detail by using probability theory, and (2) systematically evaluate the effect of the properties of soil, plant and rainfall on the stochastic erosion events by employing the LRM approach. The probabilistic description attribution approach constitutes an integrated probabilistic assessment based on event-driven probability theory and data-driven experimental observation. The investigation of stochastic soil erosion events by integrated assessment is an innovative and important complement in understanding soil erosion from the stochasticity viewpoint, and could also provide an alternative way to assess the efficacy of ecological restoration for conserving soil and water resources in a semi-arid environment.

Each observed stochastic weather condition with different durations in the field
monitoring period was defined as a random experiment. All possible outcomes
of a random experiment constituted a sample space (

The random event duration in OCIRS is an important property of stochastic weather conditions. In particular, the duration property of I events was closely related to the transmission of stochastic signals of rainfall into the R and S events. According to the rainfall duration patterns in China (Wei et al., 2007), the time interval between two adjacent I events is set to be more than 6 h, forming the criterion for individual rainfall classification. Meanwhile, based on the observation of random events over five consecutive rainy seasons, we summarised the duration property of all I events and further classified them into four mutually exclusive I event types: a random extremely long rainfall event type (Ie event, for short), a random general long-duration rainfall event type (Il event, for short), a random spanning rainfall event type (Is event, for short) whose duration spans two consecutive days, and a random within rainfall event type (Iw event, for short) occurring in a day. The C event can also be divided into two types at the daily scale: the random non-rainfall event type lasting a whole day (Cd event, for short) and the random non-rainfall event type whose duration is less than 24 h (Ch event, for short) which is interrupted by an I event.

Table 2 shows the physical, probabilistic properties and implications of all random event types in OCIRS. The classification process of all random event types is illustrated in Fig. 1a, and a Venn diagram of all random event types in OCIRS is shown in Fig. 1c. Considering the observed longest duration of an Ie event approximating 72 h, in Fig. 1b, we have summarised a series of random event sequences in terms of different combination patterns of I and C events in every 3 consecutive days during the whole monitoring period.

The OCIRS system:

Definition and explanation of all random events in OCIRS.

In the sample space

We define

The random variables

Based on the Bayes formula theory (Sheldon, 2014), if we want to evaluate
how much the probabilistic contributions of

Firstly, we constructed an event-driven logistic function, and defined

Secondly, assuming that the probabilistic distribution of

Finally, taking the natural logarithms of both sides of Eq. (16),
we transform the odds of stochastic runoff event into the linear Eq. (17)
reflecting the standard expression of LRM:

The study was implemented in the Yangjuangou Catchment (36

Study area and experimental design:

In the Yangjuangou Catchment, systematic long-term field experiments have been conducted, including the monitoring of soil erosion (Liu et al., 2012; Zhou et al., 2016), observation of soil moisture dynamic (Wang et al., 2013; Zhou et al., 2015) and assessment of soil controlling service in this typical water-restricted environment (Fu et al., 2011).

In this study, we first monitored the soil erosion events occurring in three
typical restoration vegetation types (

Secondly, we systematically monitored the hydrological properties of soil in different restoration vegetation types. In the rainy season of 2010, we began to measure the dynamics of soil moisture in the study region (Wang et al., 2013). The real-time dynamic data of soil water content at intervals of 10 min were recorded by S-SMC-M005 soil moisture probes (Decagon Devices Inc., Pullman, WA, USA), and were collected by HOBO weather station logger (Fig. 2c). These data provided the information about average antecedent soil moisture (ASM) before every rainfall event occurring in the two rainy seasons between 2010 and 2012. We further measured the field-saturated hydraulic conductivity (SHC) in all vegetation types by a model 2800 K1 Guelph permeameter (Soilmoisture Equipment Corp., Santa Barbara, CA, USA) to determine the average infiltration capability of the soil matrix (Fig. 2d).

Thirdly, we investigated the morphological properties of different
vegetation types in each runoff-plot for 2–3 times over different periods of
rainy season. We measured the average crown width, height and the thickness
of litter layer in three restoration vegetation types by setting 60

Finally, two tipping bucket rain gauges were installed outside the runoff plots to automatically record the rainfall processes over the five rainy seasons with an accuracy of 0.2 mm precipitation. Table 3 summarises the properties of the four types of random rainfall event, and the basic characteristic of soil and vegetation is shown in Table 4.

Main characteristics of the four types of random rainfall event over five rainy seasons.

Basic properties of soil, vegetation and erosion in different restoration vegetation types.

The definition and classification of properties of rainfall soil and plant ordinal variables.

We employed nonparametric statistical tests – one-way ANOVA and post hoc LSD – to determine the significant difference of soil, vegetation and erosive properties in the three restoration vegetation types. The maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) (Robert et al., 2013) were explored to compare the suitability of the binomial PMF and Poisson PMF for predicting the uncertainty of runoff and sediment generation over the long term.

The probabilistic distribution of random rainfall events (I events) and random non-rainfall events (C events) forms the environmental stochasticity which is the background of stochastic soil erosion generation. Within the OCIRS framework, the stochastic environment at monthly and seasonal scales over five rainy seasons is described in Fig. 3. For the rainy seasons of 2008 to 2012, the probability of I event generation first increased with later monitoring period and then decreased in the last two rainy seasons. In the rainy season of 2008, the average probability of I event was lower than the other four rainy seasons, being less than 15 %. However, the I event type was most complex in 2008. The random extremely long rainfall event (Ie event) only appeared in this rainy season, with the probability reaching 2.5 %. On the other hand, the average probability of I event was the highest in the rainy season of 2010, being larger than 18 %. But there were only two types of I events (Iw and Is events) in this rainy season. Over the five rainy seasons, the average probability of Iw (12.3 %) and Ie (0.8 %) event occurrence was the highest and lowest, respectively. The average probability of Is (1.7 %) and Il (1.3 %) events ranged between Iw and Ie. The probability of Cd event was higher than Ch in each month of rainy season, with average probability being 54.4 and 29.4 %, respectively. As seen in Table 3, the difference in average precipitation and duration of the four types of I events was significant. But the average rainfall intensity of Iw and Is events was nearly twice that of Il and Ie events.

The probability distribution of different random rainfall event types (Iw, Is, Il and Ie) and random non-rainfall event types (Ch and Cd) at monthly and seasonal scales from the rainy seasons of 2008 to 2012.

The stochasticity of erosion events was quantified by the probability of runoff and sediment generation in three restoration vegetation types (T1, T2 and T3) at monthly and rainy season scales (Fig. 4). Over the five rainy seasons, the probability of soil erosion occurring in all vegetation types generally decreased with later monitoring period, and then increased in 2012. At the early period of erosion monitoring (2008), the randomness of erosion events is similar, and the probability of R and S events ranged from 6 to 13 % and from 3 to 13 % respectively. After that, from the rainy seasons of 2009 to 2011, the highest probabilities of erosion events in each vegetation type all concentrated in the July and August of each season. Regarding runoff production, the average probability of R event in T1 (3.78 %) was less than that for T2 (5.60 %) and T3 (5.58 %) under the same precipitation condition. With respect to sediment yield, the average probability of S event in T1 (1.65 %) was also the lowest in all restoration vegetation types. In particular, in the last two rainy seasons, there was no S event occurring in T1, but the average probability of S event in T2 and T3 reached 1.83 and 3.36 % respectively in the corresponding rainy seasons. Consequently, affected by the same stochastic signal of rainfall events, T1 and T3 have the lowest and highest probability of erosion event generation over the five rainy seasons respectively.

The probability distribution of random runoff and sediment events occurring in three restoration vegetation types at monthly and seasonal scales from the rainy seasons of 2008 to 2012; the Arabic numbers and letter “T” on the abscissa indicate the month and season respectively (also in the following figures).

More detailed stochastic information of erosion events in different vegetation types was simulated by binomial and Poisson PMFs at monthly scale. We also compared the frequency distributions of different numbers of observed erosion events with the corresponding simulated results by the two PMFs in Fig. 5. Firstly, as to the detailed stochastic information of R events, the two PMFs generally provided a better simulation to the observations in T1 than in T2 and T3. When comparing the simulated and observed values, the binomial PMF supplied better simulation to the observed numbers of time of R events with larger frequency (such as 2–4 times) than did the Poisson PMF. However, the Poisson PMF simulated the observed numbers of time of R events with lower frequency (such as 6–8 times) better than the binomial PMF. Secondly, in relation to the detailed stochastic information of S event, the two PMFs provided better simulation to the observations in T3 than in T1 and T2. In particular, when the number of times of S event generation reaches two in T1 and T2, the corresponding simulated probability values were all nearly two times larger than the observed frequencies, reflecting the greatest simulation error of the two PMFs. Moreover, with the restoration vegetation types changing from T1 to T3, both the simulated and observed numbers of time of R and S events with largest probability or frequency increased in consistently. In summary, comparing the observed frequency of numbers of erosion events, both PMFs showed good simulation ability to detail the stochasticity of runoff and sediment events at the monthly scale.

The comparison between simulation of stochasticity of runoff and sediment events by binomial and Poisson PMFs and the observed frequencies of numbers of times of soil erosion events in three restoration vegetation types; Exp_B and Exp_P indicate the simulated values in binomial and Poisson PMF respectively; the histogram shows the observed values.

The Bayes model was applied to analyse the effect of random rainfall events (including Iw, Is, Il and Ie) on stochastic erosion events in different restoration vegetation types. Specifically, the Bayes model evaluated the different probabilistic contributions of four types of I events on one observed erosion event stochastically generated in specific vegetation type at monthly and rainy seasonal scales (Fig. 6). In the rainy season of 2008, the types of I events driving one stochastic erosion event was more complex than in the other rainy seasons. In contrast, only one stochastic soil erosion occurrence in three vegetation types was attributed to Iw and Is events in the rainy season of 2010. In the other three rainy seasons, when one R or S event stochastically generated in T1, the main contributing I event types concentrated on Is and Il events, which have relatively higher precipitation and longer duration, respectively. On the other hand, if one R or S event occurred in T2 or T3 randomly, the main contributing I event type was the Iw event which, however, had no contribution to S event occurring in T1.

The distribution of probabilistic contribution of four random rainfall event types on any one runoff or sediment event stochastically occurring in three restoration vegetation types at monthly and seasonal scales from rainy season of 2008 to 2012.

In general, over five rainy seasons, the composition of I event driving one R event was more complex than that driving one S event. The relatively longer-duration rainfall events (Il and Ie) became the main probabilistic contributors of one stochastic erosion event occurring in T1, and the relatively stronger-intensity rainfall events (Iw and Is) mainly caused one random erosion event occurring in T2 and T3.

According to the results of significant difference analysis in Table 4, we defined the properties of soil and plant as ordinal variables, and classified them into four grades (Table 5). Meanwhile, based on previous studies (Liu et al., 2012; Wei et al., 2007) and rainfall properties in this study area, we further subdivided all precipitation and rainfall intensity into four grades with different scores.

Logistic regression model to analyse the single effect of rainfall, plant and soil ordinal variable on the erosion events presence/absence in all restoration vegetation types.

Logistic regression model to analyse the interactive effect of rainfall, plant and soil ordinal variables on the erosion events presence/absence in all restoration vegetation types.

First, the intensity of positive and negative effects of a single influencing factor on the probability of runoff and sediment generation in all restoration vegetation types was quantified in terms of odds ratio of erosion events by LRM (Table 6). In the LRM, the highest and lowest odd ratios appeared in rainfall intensity ordinal variable (INT) and average crown width ordinal variable (CRO). An increasing INT and CRO (from middle to extreme grade) significantly increased and decreased the odds ratio of erosion events, respectively. This means that INT and CRO have two of the most important roles in improving and restraining the probability of stochastic erosion generation in all restoration vegetation types. Additionally, the increasing of antecedent soil moisture ordinal variable (ASM) and the SHC ordinal variable (from middle to high grade) in the LRM also significantly increased and decreased the odds ratio of R and S events, respectively. However, the average thickness of litter layers (TLL) ordinal variable did not have significant effect on the odds ratio of erosion events. Tables S1 and S2 in the Supplement systematically describe the processes of LRM to evaluate the effect of single factors on the odds ratio of erosion event.

Secondly, we applied LRM to evaluate the interactive effects of multiple influencing factors on the odds ratio of R and S events in all restoration vegetation types (Table 7). Regarding the interactive effect of two soil hydrological properties, the interaction between low grade of SHC and increasing grade of ASM significantly raised the odds ratio of erosion events – the odds ratio of R and S events affected by the interactive effects of low-grade SHC and extreme-grade ASM were respectively 7.02 and 1.82 times larger than the interactive effects of low-grade SHC and low-grade ASM. Similarly, regarding the effect of two vegetation properties, the interactive effect of low-grade CRO and increasing-grade TLL reduces the odds ratio of erosion events – the odds ratio of R and S events influenced by the interaction between low-grade CRO and high-grade TLL were respectively only 0.12 and 0.33 times larger than the interactive effects of low-grade CRO and low-grade TLL. Additionally, with respect to the interaction between soil and plant properties, the interactive effect of low-grade CRO and increasing-grade ASM properties also significantly raised the odds ratio of erosion events. The processes of LRM used to evaluate the interactive effect of multiple factors on odds ratio of erosion event are detailed in Supplement Tables S3–S5.

The probabilistic attribution and description of stochastic erosion events constituted the framework of integrated probabilistic assessment (IPA).

First, as one pattern of probabilistic attribution in the IPA, the Bayes
model supplies a supplementary view and algorithm about how to evaluate the
feedback of a result which had stochastically occurred on all possible
reasons (Wei and Zhang, 2013). Under the conditions of insufficient
information about an occurred result, the Bayes model can determine which
reasons have relatively greater probability to trigger the occurrence of
the result through some prior information. Specific to this study, the Bayes
model was used to evaluate the probabilistic contribution of four types of I
events on one stochastic R (P

Secondly, as a pattern of probabilistic description in the IPA, the binomial and
Poisson PMFs are two crucial probabilistic functions to characterise many
random hydrological phenomena and to model their ecohydrological effects in
natural condition (Eagleson, 1978; Rodriguez-Iturbe et al., 1999, 2001). In
this study, the two PMFs were found to give good simulations of the
frequency of times of soil erosion events in three restoration vegetation
types. However, it is necessary and meaningful for the reliability and
accuracy of the IPA to assume whether the two PMFs can both stably and
reasonably simulate the erosion stochasticity at closed-runoff plot over a
longer monitoring period. Therefore, based on the above assumption, two
important point estimation methods – the maximum likelihood estimator (MLE)
and uniformly minimum variance unbiased estimator (UMVUE) (Robert et al.,
2013) – were applied to evaluate the stability of erosion stochasticity
estimation by means of analysing the unbiasedness and consistency of

Thirdly, besides having better simulation of the stochastic soil erosion events at larger temporal scale, the Poisson PMF is also more suitable for simulating the randomness of S event in the closed-design plot system than the binomial PMF.

Following the hypothesis of Boix-Fayos et al. (2006), the closed runoff-plot design
forms an obstruction to prevent the transportable material from entering the
close monitoring system, which, in particular, leads the transport-limited
erosion pattern to gradually transform into a detachment-limited pattern in
the closed plot over time (Boix-Fayos et al., 2007; Cammerraat, 2002).
Consequently, with the extension of monitoring period, this closed-runoff plot design would make it more and more difficult for the sediment to
migrate out of the plot, which also reduces the probability of observed S events
under the same precipitation condition. In fact, the effect of closed-runoff plot on stochastic sediment event is also implied
by the algorithm of the Poisson PMF. Specifically, in order to satisfy that

The effects of rainfall, soil and vegetation properties on erosion stochasticity in different restoration vegetation types were evaluated by LRM. This integrated stochastic rainfall events with their precipitation and intensity grades, and connected the ecohydrological functions of soil and plant with their classified hydrological and morphological features.

Just as in previous studies (Verheyen and Hermy, 2001a, b; Verheyen et al., 2003), LRM in this study explored the relative importance of morphological features disturbing the transmission of stochastic signal of I events into R and S events in different restoration vegetation types. These disturbances are closed related to the complex hydrological functions owned by different morphological structures, which finally affect the whole processes of runoff production and sediment yield (Bautista et al., 2007; Puigdefábregas, 2005).

First, many previous field experiments and mechanism models have shown that canopy structure has capacity for intercepting precipitation. This specific hydrological function can prevent rainfall from directly forming overland flow or splashing soil surface particles (Liu, 2001; Mohammad and Adam, 2010; Morgan, 2001; Wang et al., 2012). The precipitation retention by canopy structure has been regarded as an indispensable positive factor to reduce the soil erosion rate. Meanwhile, as a crucial complement to understanding the hydrological function of canopy structure, the result of LRM in this study indicated that the higher-grade canopy structure was a most important morphological feature to reduce the odds ratio of random soil events in all restoration vegetation types. This result suggests that larger canopy diameter would have relatively stronger capacity for disturbing the transmission processes of stochastic signal of rainfall on the soil surface than other morphological properties. From the perspective of erosion stochasticity, the higher-grade canopy structure could finally be attributed to the lower probability of R and S event generation. Therefore, the diversity of canopy structures in different vegetation types could play a key role in reducing both the intensity and probability of soil erosion generation.

Secondly, many studies have also discovered that denser root system distribution in the soil matrix improves the overland reinfiltration (Gyssels et al., 2005). This reinfiltration process is an effective way to recharge soil water stores when the overland flow starts to occur in hillslopes, which is also an indispensable contributing factor to reduce the unit area runoff production (Moreno-de las Heras et al., 2009, 2010). In this study, the potential reinfiltration capacity of the soil matrix could be positively affected by the saturated hydraulic soil conductivity (SHC) index. Figure 7 indicates the distribution patterns of root system in three restoration vegetation types. Meanwhile, the result of LRM also implied that the grade of SHC could negatively affect the odds ratio of stochastic erosion event, which improved the understanding of the hydrological function of plant root distribution from the viewpoint of erosion randomness. This suggests that the denser root system creates more macropores in the subsurface to provide more probability of reinfiltration of overland flow. This disturbance of overland flow by SHC can reduce the probability of erosion event generation.

Morphological properties of three restoration vegetation types including the thickness of litter layer and the distribution of root system. The dashed lines indicate the diameter and depth of soil samples, approximately 10 and 30 cm respectively.

Thirdly, the litter layer was shown to play multiple roles in conserving the rainfall, by improving infiltration of throughfall, as well as cushioning the splashing of raindrops (Gyssels et al., 2005; Munoz-Robles et al., 2011; Geißler et al., 2012). Therefore, the thicker litter layer in T2 (Fig. 7) probably has stronger capacity for conserving and infiltrating throughfall, as well as inhibiting splash erosion than that of other restoration vegetation types (Woods and Balfour, 2010). Although the result of LRM indicated that there was no significant correlation between the TLL and the odds ratio of soil erosion (Table 6), the interactive effect of TLL and CRO significantly affects the odds ratio of stochastic erosion events (Table 7). The interaction result implied that, under the relative low-grade CRO condition, the higher-grade TLL could have stronger disturbance on the transmission of stochastic signals of rainfall to improve the throughfall absorption to reduce the probability of splash or sheet erosion occurrence.

Additionally, Table 7 explored more interactive effects of the soil and plant properties on the odds ratio of random runoff and sediment event. These explorations suggested that the interactions between soil and vegetation properties formed more complex hydrological functions to affect the stochastic soil erosion event during ecohydrological processes in semi-arid environment (Ludwig et al., 2005).

Although the hydrological traits of vegetation played core roles in reducing the soil erosion depending on the mechanical properties of their morphological structures (Zhu et al., 2015), the LRM analysis in this study further illuminated that these hydrological-trait morphological structures of vegetation may also play an important role in affecting the stochasticity of soil erosion. Actually, the different stochasticity of soil erosion in three restoration vegetation types reflected the different extent of disturbance of vegetation type on the transmission of stochastic signals of rainfall into soil–plant systems. Therefore, the relatively smaller canopy structure, thinner litter layer and shallower root system in T3 have relatively weaker capacity to disturb the stochastic signal of rainfall than that of T1 and T2 with obvious hydrological-trait morphological structures (Fig. 7). The effect of diverse morphological structures on stochasticity of soil erosion was a meaningful complement to studying the hydrological functions of restoration vegetation types in semi-arid environment.

The IPA is an important complement to expand on the understanding of hydrological function existing in vegetation types. The hydrological-trait of morphological structures owned by different plants is closely related to the function of vegetation-driven spatial heterogeneity (VDSH) in affecting the intensity of erosion events. The VDSH theory (Puigdefábregas, 2005) can be regarded as a clear and concise summary to emphasise the dominant role of vegetation in restructuring soil erosion processes. It reflects the effect of spatial distribution patterns of vegetation on their corresponding hydrological functions in controlling erosion rate in patch, stand and even at regional . Therefore, VDSH theory has provided an innovative view to investigating the soil erosion and other ecohydrological phenomena affected by vegetation (Sanchez and Puigdefábregas, 1994; Puigdefábregas, 1998; Boer and Puigdefábregas, 2005). In the study, depending on the long-term experimental data and fundamental probability theories, the IPA concentrated on the hydrological function of VDSH in affecting the randomness of erosion events rather than the erosion rate. This can enrich the comprehension of the hydrological function of vegetation morphological structure in soil erosion phenomena, and also be an effective complement to the application of VDSH theory in interpreting stochastic erosion events.

Additionally, in our study, the IPA also provides a new framework for practitioners to develop restoration strategies focused on controlling the risk of erosion generation rather than only on reducing erosion rate. The framework contains three stages: construction of stochastic environment, description of random erosion events, and evaluation of probabilistic attribution (Fig. 8).

The framework of integrated probabilistic assessment for soil erosion monitoring and restoration strategies.

The first stage in the framework aims to build a unified platform to describe the stochasticity of different hydrological phenomena closely related to the erosion event. This stage generally investigates the stochastic background under which soil erosion occurs, which is also an indispensable precondition for quantifying the probability of R and S in stage II. The second stage is designed to construct a phased adjustment of monitoring processes based on the principle of Bayes theory as well as on the parameter analysis of binomial and Poisson models. In this phased-adjustment monitoring, the Bayes, binomial and Poisson models were applied to simulate the randomness of erosion events in short-term, mid-term and long-term monitoring stages, respectively. This model-driven monitoring approach can be regarded as a more reasonable method to explore the complexity of stochastic erosion events in larger temporal scales, but also provide a new perspective for researchers to more effectively evaluate the stochasticity of erosion events in stage III. The objective of stage III is to assess the probabilistic attribution of rainfall, soil and vegetation properties on erosion event generation. This probabilistic attribution evaluation by LRM could develop the restoration strategies for more effectively selecting vegetation types with stronger capacity for reducing the erosion risk, and finally improve the management of soil and water conservation in a semi-arid environment.

As a result, this stochasticity-based restoration strategy was developed by a combination of experimental data with multiple probabilistic theories to deal with the soil erosion randomness under complex stochastic environment. It is different from the trait-based restoration scheme derived from the functional diversity of vegetation community to reduce the soil erosion rate (Zhu et al., 2015; Baetas et al., 2009). Meanwhile, with increased monitoring duration, more stochastic information of erosion events could be added into the IPA framework. This addition could finally fulfil the self-renewal and self-adjustment of the IPA to improve the restoration strategy for selecting more reasonable vegetation types with stronger capacity for controlling erosion risk in the long term. Therefore, the IPA framework containing three stages could translate the event-driven erosion stochasticity into restoration strategies concentrating on erosion randomness, which may be a helpful complement for restoration management in a semi-arid environment.

In this study, we applied an integrated probabilistic assessment (IPA) to describe, simulate and evaluate the stochasticity of soil erosion in three restoration vegetation types in the Loess Plateau of China, and draw the following conclusions.

In the IPA, the OCIRS was an innovative event-driven system to standardise the definition of hydrological random events, which is also a foundation for quantifying the stochasticity of soil erosion events under complex environmental conditions.

Both binomial and Poisson PMFs in the IPA can simulate the probability distribution of the numbers of runoff and sediment events in all restoration vegetation types. However, the Poisson PMF more effectively simulated the stochasticity of soil erosion at larger temporal scales.

The difference of morphological structures in restoration vegetation types is the main source of different stochasticity of soil erosion from T1 to T3 under the same rainfall condition. Larger canopy, thicker litter layer and denser root distribution could more effectively affect the transmission of stochastic signal of rainfall into soil erosion.

The IPA is an important complement to developing restoration strategies to improve the understanding of stochasticity of erosion generation rather than only of the intensity of erosion event. It could also be meaningful to researchers and practitioners to evaluate the efficacy of soil control practices in a semi-arid environment.

All the data used in this study are available on request, and they can be accessed by contacting the corresponding author.

Let

Let the random sample

Therefore,

Let

Let the random sample

Let

Firstly, MLE

The authors declare that they have no conflict of interest.

This work was funded by the National Natural Science Foundation of China (no. 41390464) and the National Key Research and Development Program (no. 2016YFC0501602). We are especially grateful to the associate editor and the two reviewers, whose suggestions and advice improved the quality of this study. We also thank Chen Lin-An with National Chiao Tung University (NCTU) for his great help on the mathematical statistical inference in this paper, as well as Liu Yu, Liu Jianbo and Wang Jian for their support for soil erosion monitoring. Edited by: L. Wang Reviewed by: J. P. Puigdefábregas, C. Miao, and one anonymous referee