Introduction
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.
aStochasticity (uncertainty)
bApproach or method
cDriven types
Main hydrological behaviours
Main influencing factors
Spatiotemporal scale
Reference
Stochasticity of soil erosion rate
Runoff connectivity
Probabilistic model Conceptual model
1. Data-mapping 2. Theory
Infiltration processes Precipitation
Topography Soil depth
Hillslope scale in the USA
Janzen and McDonnell (2015)
Runoff processes
Probabilistic model Conceptual model
1. Simulation 2. Theory
Infiltration processes Precipitation
Topography
Janzen and McDonnell (2015)
Runoff production
Probabilistic model Conceptual model
1. Theory 2. Simulation
Runoff absorption Water storage Infiltration capacity
Soil moisture Evaporation recharge
Point and basin scale
Moore (2007)
Flood prediction and runoff
Probabilistic model Multivariate analysis
1. Simulation 2. Data calibration
Stochastic rainfall process
Parameters in rainfall–runoff model
Multiple catchment scales in Iran
Yazdi et al. (2014)
Rainfall and runoff processes
Probabilistic model hydrological mechanism
1. Simulation 2. Random event 3. Theory
Soil storage
Given climate regime hydraulic conductivity landform development
Hillslope scale
Freeze (1980)
Erosion rate
Probabilistic model Mechanical mechanism
1. Data calculation 2. Stochastic forcing
Bed shear stress Critical shear stress
Laboratory scales in the Netherlands
Prooijen and Winterwerp (2010)
Erosion rate
Physical model Probabilistic model Conceptual model
1. Theory 2. Simulation
Simulated near-bed flow
Soil structure Oscillating flow
Sidorchuk (2005)
Erosion risk
Empirical model Geostatistics
1. Data mapping
Erosive precipitation
Factors in RUSLE
Annual and regional scales in China
Jiang et al. (2012)
Uncertainty of soil loss
Empirical model Geostatistics Error analysis
1. Simulation 2. Data calibration
Erosive precipitation Runoff and sediment
Spatiotemporal rainfall erosivity distribution
Annual time and catchment scale in the USA
Wang et al., 2002
Uncertainty and variability of erosion rate
Empirical model
1. Hypotheses 2. Data calculation
Total rainfall volume and 30 min rainfall intensity
Stochastic environment conditions Scale effect
Kim et al. (2016)
Stochasticity of factors affecting soil erosion rate
Soil moisture related to soil erosion
Probabilistic model Physical model
1. Hypotheses, 2. Simulation 3. Theory
Precipitation Evapotranspiration
Temporal patterns of rainfall property
Daily time and hillslope scale in
Ridolfi et al. (2003)
Antecedent soil moisture related to soil erosion
Probabilistic model Physical model
1. Data mapping 2. Theory
Runoff response Infiltration processes
Daily time and multiple catchment scales in Spain
Castillo et al. (2003)
Stochastic rainfall related to flood and runoff
Probabilistic model Conceptual model
1. Data calibration 2. Random event 3. Hypothesis
Stochastic storm Runoff and flood
Parameters in Peak flow models
Hourly–daily time and multiple catchment scales in Germany
Haberlandt and Radtke (2014)
Stochastic rainfall related to runoff and erosion
Physical model Empirical model
1. Simulation 2. Data calibration
Overland/channel flow Erosion transport Precipitation
Spatiotemporal rainfall distribution
Seasonal and annual time catchment scale in the USA
Lopes (1996)
Uncertainty of soil erodibility
Empirical model Geostatistics
1. Simulation 2. Data mapping
Spatiotemporal soil types, depth and parent material
Regional scales in the USA
Wang et al. (2001)
Stochastic rainfall related to runoff
Probabilistic model Conceptual model Physical model
1. Data calibration 2. Theory
Sewer overflows
Rainfall depth and duration, climate conditions
Seasonal and annual time catchment scales in Spain
Andrés-Doménech et al. (2010)
a The main contents of different studies focusing on
the stochasticity (uncertainty) of soil erosion and its influencing factors.
b The main statistical methods or different
types of mathematical and physical models employed to describe and analyse the stochasticity
of soil erosion. c The main properties of analysis framework in the different studies and
the characteristics of data application on the evaluation of stochasticity of soil
erosion.
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.
Method
Definition and classification of random events
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 (Ω) defined as a
random observational event (O event, for short). Two mutually exclusive random
event types – random rainfall event (I event, for short) and random
non-rainfall event (C event, for short) – constituted the O event.
Precipitation is a necessary condition of runoff generation, and the random
runoff production event (R event, for short) is a subset of the I event.
Similarly, R event is also a necessary condition of random sediment migration
event (S event, for short), which leads to S event being a subset of R event. As a
result, O, C, I, R and S events constituted a random events framework
(OCIRS) to reflect the stochasticity of the environment in which soil erosion
events occur.
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: (a) a flow chart to
determine random event types in the OCIRS framework; (b) the
different combination patterns of rainfall and non-rainfall events in 3
consecutive days to form ten observed random event sequences in five rainy
seasons; (c) Venn diagram showing the relationship between
random event types in the OCIRS framework.
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Definition and explanation of all random events in OCIRS.
Symbol
Physical meaning of random event types
Probabilistic meaning of random event types
Influencing factors and implication
O
observation events with time step ranging from 0 to 72 h, including non-rainfall and rainfall events
random events composing the sample space of OCIRS system. The probability P(O) = 1
indicating the general stochastic weather conditions over rainy seasons
C
non-rainfall events with time step ranging from 0 to 24 h, including sunny or cloudy weather condition at hour or day scales
random events, the probability of C events is the ratio of numbers of C events to O events C ⊂ O, 0 ≤ P(C) ≤ P(O) = 1
implying the extent of evaporation or potential evapotranspiration in weather condition
Cd
non-rainfall events with time step being 24 h, including observed sunny or cloudy at day scale
random events composing the subset of C events, Cd ⊆ C, 0 ≤ P(Cd) ≤ P(C)
implying the duration of evaporation or evapotranspiration at day scale
Ch
non-rainfall events with time step being less than 24 h, including observed sunny or cloudy at hour scales which intercepted by rainfall events within a day
random events composing the subset of C events, the intersection of Ch and Cd is null, Ch ⊆ C, Cd ∪ Ch = C, Cd ∩ Ch =∅, 0 ≤ P(Ch) ≤ P(C)
influenced by the frequency of rainfall events generation, and implying the alternation of sunny and rainy in a day
I
an individual rainfall event with different precipitation, intensity and duration ranging from 0 to 72 h, the time interval between two I events is more than 6 h
random events, the probability of I event is ratio of numbers of I events to O events over observation I ⊂ O, I ∪ C = O, I ∩ C = ∅, 0 ≤ P(I) ≤ P(O) = 1
a driven force of soil erosion, which could be intercepted by vegetation and transformed into throughfall
Ie
an extreme longest individual rainfall event whose average precipitation, intensity and duration were 96.6 mm, 0.022 mm min-1, and 73 h, respectively.
random events composing the subset of I events, Ie ⊆ I, 0 ≤ P(Ie) ≤ P(I)
rainfall events with low intensity and longest duration, leading to infiltration–excess runoff generation
Il
second longest individual rainfall event type whose average precipitation, intensity and duration were 47.3 mm, 0.027 mm min-1, and 30 h, respectively.
random events composing the subset of I events, the intersection of Il and Ie is null, Il ⊆ I, Il ∩ Ie = ∅, 0 ≤ P(Il) ≤ P(I)
rainfall events with low intensity and long duration, leading to infiltration–excess runoff generation
Is
rainfall event type spanning 2 days whose average precipitation, intensity and duration were 22.7 mm, 0.042 mm min-1, and 10 h, respectively
random events composing the subset of I events, Is ⊆ I, Is ∩ Il ∩ Ie = ∅, 0 ≤ P(Il) ≤ P(I)
rainfall events with strongest rainfall intensity in middle duration, leading to runoff and sediment generation
Iw
rainfall event type occurring within a day whose average precipitation, intensity and duration were 9.8 mm, 0.045 mm min-1, and 5 h, respectively. It usually occurs several times within 1 day.
random events composing the subset of I events, Iw ⊆ I, Iw ∩ Is ∩ Il ∩ Ie = ∅, Iw ∪ Is ∪ Il ∪ Ie = I, 0 ≤ P(Iw) ≤ P(I)
rainfall events with least and shortest precipitation and duration, which is difficult to trigger soil erosion
R
runoff event type occurring on vegetation land type; it occurs on rainfall processes, and its duration is negligible
random events responding to I events, R ⊂ I, R ∩ C = ∅, 0 ≤ P(R) < P(I)
influenced by rainfall and vegetation properties
S
sediment event occurring on vegetation land types, it occurs on runoff processes, and its duration is negligible
random events responding to R events, S ⊂ R ⊂ I, S ∩ C = ∅, 0 ≤ P(S) ≤ P(R) < P(I)
driven by R events, and affected by rainfall and vegetation properties
Probabilistic description of erosion event
Conditional probability of erosion event
In the sample space Ω, for any random event type E in
OCIRS, we defined P(E) as the proportion of time that E
occurs in terms of relative frequency:
PE=limn∞nEn=pE,pE∈[0,1].
Theoretically, nE is the number of times in n
outcomes of observed random experiment that the event E occurs.
According to the law of total probability (Robert et al., 2013), the
probability of R event is defined as
PR=PRI=PR|∪m=14ImP∪m=14Im=∑m=14PR|ImPIm=pR.
Im, m= 1, 2, 3 and 4 represent the Ie, Il, Is, and Iw
respectively, and PR|Im represents the conditional probability that R event occurs given that the
mth I event type has occurred. Similarly, the probability of S
event is defined as
PS=PSI=PS|∪m=14ImP∪m=14Im=∑m=14PS|ImPIm=pS.
Equations (2) and (3) quantify the stochastic soil erosion events by using
conditional probability.
Probability distribution functions of erosion event
We define X and Y as two discrete random variables, representing two
real-valued functions defined on the sample space (Ω). Let X and Y
denote the numbers of times of R and S event occurrence respectively, and
assign the sample space Ω to another random variable Z. XR=x,YS=yZΩ=z,y≤x≤z. The x, y, and z are integers. The ranges of X and Y are
RX=allx:x=XR,allR∈Ω and RY=ally:y=YS,allS∈Ω. The probability of xi or yj numbers of
times of R or S events can be quantified by probability mass function (PMF)
as follows:
PMFXxi=PRi:XRi=xi,xi∈RXPMFYyj=PSj:YSj=yj,yj∈RYfori≥j.
The PMF in Eqs. (4) and (5) describe the general expression of probability
distribution of all possible numbers of times of R or S events.
The random variables X and Y obey the binomial distribution with n
independent Bernoulli experiments (Robert et al., 2013). Therefore, the PMF
of X and Y can be defined as follows:
PMFXbinx=PXbinX=x=nxpRx1-pRn-xx=0,1,2,…,n0elsewherePMFYbiny=PybinY=y=nypSy1-pSn-yy=0,1,2,…,n0elsewhere,
where x and y indicate all possible numbers of times of R
and S occurring over n I events. However, when the Bernoulli
experiment is performed infinite independent times (n→∞),
the binomial PMF can be transformed into the Poisson PMF (proved in Appendix A), and finally expressed as follows:
PMFXpoi(x)=PXpoi(X=x)=λRxe-λRx!x=0,1,2,…0elsewherePMFYpoi(y)=PYpoi(Y=y)=λSye-λSy!y=0,1,2,…0elsewhere,
where the parameter λR≈npR,λS≈npS.
Equations (6)–(9) reflect two
PMF models to simulate the probability distribution of R or S events.
Probabilistic attribution of erosion events
Bayes model
Based on the Bayes formula theory (Sheldon, 2014), if we want to evaluate
how much the probabilistic contributions of kth type of random
rainfall event on one stochastic runoff or sediment event which has been
generated and observed, the Bayes model can quantify the results as follows:
PIk|R=P(IkR)P(R)=PR|IkP(Ik)∑m=14PR|ImPImPIk|S=P(IkS)P(S)=PS|IkP(Ik)∑m=14PS|ImPIm.
In fact, the Bayes model provides an important explanation that how the a
priori stochastic information (PIk) was
modified by the posterior stochastic information (PR or PS). The application of Bayes model in
Eqs. (10)–(11) reflects the feedback of random erosion events on
the stochastic rainfall events. It could also be regarded as one pattern of
probabilistic attribution to assess the effect of different random rainfall
events on the uncertainty of soil erosion events without considering the
diversity of restoration vegetation.
Logistic regression model
Firstly, we constructed an event-driven logistic function, and defined
YR and YS as two dichotomous dependent
variables. When we denote YR and YS as 1 or 0, it means that a R and S event has occurred or not occurred.
Given that YR is a dichotomous dependent variable of R event
in the linear probability model in can be expressed as follows:
YRi=α+β1x1i+β2x2i+⋯+βnxni+ξi=α+∑n=1nβnxni+ξi.
We then further transform Eq. (12) into the conditional probability of R
event which has occurred in the ith observation time as follows:
PYRi=1|∩n=1nxni=Pα+∑n=1nβnxni+ξi≥0=Pξi≤α+∑n=1nβnxni=Fα+∑n=1nβnxni,
where αβ are constants and Fα+∑n=1nβnxni is the cumulative distribution function of ξi when ξi=α+∑n=1nβnxni. Equations (12) and (13) quantify the stochasticity of
YRi depending on the linear combination of n influencing
factors xn and measurement error ξ under
i observation times of stochastic runoff generation.
Secondly, assuming that the probabilistic distribution of ξi
satisfies logistic distribution and PYRi=1|∩n=1nxni=pi, then the LRM expression of
YRi=1 is deduced as follows:
pi=Fα+∑n=1nβnxni=11+e-α+∑n=1nβnxni=eα+∑n=1nβnxni1+eα+∑n=1nβnxni.
Correspondingly, the LRM of YRi=0 can be expressed as
PYRi=0|∩n=1nxni=1-pi=11+eα+∑n=1nβnxni.
The ratio of Eq. (14) to (15) is defined as the odds of the R event:
Odds=pi1-pi=eα+∑n=1nβnxni1+eα+∑n=1nβnxni11+eα+∑n=1nβnxni=eα+∑n=1nβnxni,odds∈0,1.
In this study, odds in Eq. (16) is a probabilistic attribution
index to quantify how much the n influencing factors affect the
generation of the ith stochastic runoff event. Specifically, when
the odds of an influencing factor is greater (less) than 1, it means that
the corresponding influencing factor exerts positive (negative) effects
on the probability of R generation.
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:
lnPYRi=1|∩n=1nxniPYRi=0|∩n=1nxni=lnpi1-pi=α+∑n=1nβnxni.
LRM can be regarded as another probabilistic attribution pattern to
evaluate the effect of multiple impacting factors – such as soil, vegetation
and rainfall – on the randomness of soil erosion events occurring in
different restoration vegetation types.
Experimental design and data analysis
Study area
The study was implemented in the Yangjuangou Catchment (36∘42′ N,
109∘31′ E; 2.02 km2), which is located in the
typical hilly-gully region of the Loess Plateau in China (Fig. 2a). A
semi-arid climate in this area is mainly affected by the North China
monsoon. Annual average precipitation reaches approximately 533 mm, and the
rainy season here spans from June to September (Liu et al., 2012). The
Calcaric Cambisol soil type (FAO-UNESCO, 1974) with weak structure and
higher erodibility in the Loess Plateau is vulnerable to water erosion. For
these reasons, soil and water loss was one of the most serious environmental problems to
degrade the ecosystem in the Loess Plateau before the 1980s (Miao et
al., 2010; Wang et al., 2015). After that, as a crucial soil and water
resource protection project, the Grain-for-Green Project was widely
implemented in the Loess Plateau. A large number of steeply sloped croplands
were abandoned, restored or naturally recovered by local shrub and herbaceous
plants (Cao et al., 2009; Jiao et al., 1999). In the Yangjuangou Catchment,
the main restoration vegetation distributed on hillslopes includes
Robinia pseudoacacia Linn., Lespedeza davurica,
Aspicilia fruticosa, Armeniaca sibirica, Spiraea pubescens and
Artemisia copria. All the restoration vegetation was planted
over 20 years ago.
Study area and experimental design: (a) location of the Yangjuangou
Catchment; (b) three restoration vegetation types including
Armeniaca sibirica (T1), Spiraea pubescens (T2) and
Artemisia copria (T3); (c) the dynamic measurement of soil moisture
and data collection to provide the information about average antecedent soil
moisture; (d) the measurement of field-saturated hydraulic conductivity to
determine the average infiltration capability; (e) investigation of the
morphological properties of restoration vegetation by setting quadrats.
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Design and monitoring
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 (Armeniaca sibirica (T1),
Spiraea pubescens (T2) and Artemisia copria (T3)) from the
rainy seasons of 2008–2012 (Fig. 2b). Each restoration vegetation type
was designated by three 3 m by 10 m closed-runoff plots located on
southwest-facing hillslopes with 26.8 % aspect. The boundaries of each
runoff plot were perpendicularly fenced with impervious polyvinyl chloride
(PVC) sheet of 50 cm depth. Collection troughs and storage buckets were
installed at the bottom boundary to collect the runoff and sediment (Zhou et
al., 2016). Under natural precipitation condition, we recorded the number of
times that stochastic runoff and sediment events occurred in each
runoff plot over the five rainy seasons. Also, we collected runoff and
sediment, separated them and, after settling for 24 h, the samples were
dried at 105∘ over 8 h and weighed.
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 × 60 cm
quadrats in each runoff plot (Bonham, 1989) (Fig. 2e).
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.
Rainy
Rainfall
Average
Average
Average
season
event
precipitation
intensity
duration
types
(mm)
(mm min-1)
(h)
2008
Iw
16.7
0.122
2.3
Is
19.2
0.066
4.8
Il
53.2
0.032
27.7
Ie
96.6
0.022
73.2
2009
Iw
9.0
0.027
5.6
Is
35.4
0.059
10.0
Il
47.9
0.032
24.9
Ie
×
×
×
2010
Iw
9.0
0.018
8.3
Is
7.6
0.012
10.6
Il
×
×
×
Ie
×
×
×
2011
Iw
3.3
0.031
1.8
Is
21.5
0.040
9.0
Il
42.5
0.020
35.4
Ie
×
×
×
2012
Iw
10.8
0.028
6.4
Is
30.0
0.031
16.1
Il
45.5
0.023
33.0
Ie
×
×
×
Average
Iw
9.8
0.045
4.9
Is
22.7
0.042
10.1
Il
47.3
0.027
30.3
Ie
96.6
0.022
73.2
Basic properties of soil, vegetation and erosion in different
restoration vegetation types.
Basic properties of different
hN
Restoration vegetation
vegetation type
types
Armeniaca sibirica
Spiraea pubescens
Artemisia copria
Type 1 (T1)
Type 2 (T2)
Type 3 (T3)
Topography property
Slope aspect
9
Southwest
Southwest
Southwest
Slope gradation (%)
9
≈26.8
≈26.8
≈26.8
Slope size for each (m)
9
3 × 10
3 × 10
3 × 10
Soil property
aDBD (g cm3)
30
1.28 ± 0.08
1.16 ± 0.12
1.23 ± 0.10
Clay (%)
30
11.07 ± 2.43
11.98 ± 3.05
9.54 ± 1.48
Silt (%)
30
26.11 ± 1.50
25.24 ± 3.84
26.72 ± 2.87
Sand (%)
30
62.82 ± 0.94
62.78 ± 4.51
63.74 ± 3.24
bTexture type
Sandy loam
Sandy loam
Sandy loam
cSHC (cm min-1)
20
0.46 ± 0.82(a)
2.22 ± 0.66(b)
0.50 ± 0.60(a)
dSOM ( %)
30
1.28 ± 0.63(a)
0.98 ± 0.15(b)
0.90 ± 0.09(b)
Vegetation property
Restoration years
9
20
20
20
Crown diameters (cm)
27
211.6 ± 15.4(c)
80.5 ± 4.5(b)
64.1 ± 6.3(a)
Litter layer (cm)
30
1.2 ± 0.3(a)
3.4 ± 1.8(b)
1.8 ± 0.5(a)
Height (cm)
27
256.3 ± 11.1(c)
128.3 ± 8.3(b)
61.8 ± 1.1(a)
LAI
27
×
2.31
1.78
eAve. coverage (%)
27
85
90
90
Rainfall/erosion property
Times of rainfall events
130
Times of runoff events
30/30/30
45/45/45
45/45/45
Times of sediment events
13/13/13
19/19/19
32/32/32
fAve. runoff depth (cm)
0.012(a)
0.014(a)
0.083(b)
gAve. sediment amount (g)
5.8(a)
6.8(a)
25.7(b)
a Dry bulk density. b Texture type is
determined by textural triangle method based on USDA. c Field saturated
hydraulic conductivity, and all the values with same letter in each row
indicates non-significant difference at α= 0.05
which is the same as follow rows. d Soil organic matter. e Average coverage of
three restoration vegetation types over five rainy seasons. f Average runoff depth
in restoration types over rainy seasons. g Average sediment yield in restoration
types over rainy seasons. h Sample number.
The definition and classification of properties of rainfall soil and
plant ordinal variables.
Ordinal variable
Physical meaning of classified influencing factors
Standard of influencing factor
classification
Low
Middle
High
Extreme
(L)
(M)
(H)
(E)
PREC
classified precipitation variable of a single random rainfall event
0–15 mm
15–30 mm
30–60 mm
> 60 mm
INT
classified intensity variable of a single random rainfall event
0–0.025 mm min-1
0.025–0.05 mm min-1
0.05–0.1 mm min-1
> 0.1 mm min-1
ASM
classified variable of the antecedent soil moisture
0–5 %
5–10 %
10–20 %
> 20 %
SHC
classified variable of the filed saturated hydraulic conductivity
0–1 cm min-1
×
> 1 cm min-1
×
CRO
classified variable of the average crown width in vegetation type
0–60 cm
60–80 cm
> 80 cm
×
TLL
classified variable of the average thickness of litter layers
0–2 cm
×
> 2 cm
×
YR
dichotomous dependent variable to indicate whether a random
If YR= 1, it means that a random runoff event has occurred;
runoff event has occurred or not
If YR= 0, it means that a random runoff event has not occurred
YS
dichotomous dependent variable to indicate whether a random
If YS= 1, it means that a random sediment event has occurred;
sediment event has occurred or not
If YS= 0, it means that a random sediment event has not occurred
Statistics
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.
Discussion
The integrated probabilistic assessment of erosion
stochasticity
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 (PIk|R) and S (PIk|S) event
generated in each restoration vegetation type. Although there was no further
specific information about a stochastic soil erosion event, the prior
information (PR|ImPS|ImPIm) can provide assistance for us to assess the feedback of the stochasticity
of soil erosion on different random rainfall events by the Bayes model.
Meanwhile, (PIk|R) and
(PIk|S) also reflect the
different probability threshold values of four rainfall event types
triggering soil erosion. The Bayes model integrated with total probability
theory to systematically quantify the interactive relationship between the
stochasticity of precipitation and soil erosion, forming a relatively simple
and practical risk assessment of soil erosion event occurring in complex
restoration vegetation conditions.
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
pR, pS, λR and λS. Taking parameter analysis of
random runoff event for example, we defined Xi as the number
of times R event occurred in some specific restoration vegetation type in the
ith rainy season (i= 1, 2, 3, 4 and 5). The five
independent and identical (iid) random variables satisfy the same
and mutually independent binomial or Poisson PMFs as follows:
X1,X2,…,X5⟶iidbinomialpRorX1,X2,…,X5⟶iidPoissonλR.
Considering longer monitoring periods, we supposed that the numbers of
corresponding I events (n) and rainy seasons (i) would
approach infinity (n, i→∞), and Eq. (18) can be
transformed as follows:
X1,X2,…,Xi⟶iidbinomial(p)orX1,X2,…,Xi⟶iidPoissonλ.
We take MLE and UMVUE methods to search for the best reasonable population
estimators p^ and λ^ to approximate the unknown p and
λ in Eq. (19), and finally obtain more comprehensive stochastic
information about the randomness of R event over i rainy seasons.
Appendix B shows that the best estimator p^ in binomial PMF is
the unbiasedness and consistency of the MLE of p. However, as shown in
Appendix C, the best estimator λ^ in the Poisson PMF has more
reliability as it has not only the unbiasedness and consistency of the MLE of
λ, but also the UMVUE of MLE. The UMVUE in the Poisson PMF implies that the
lowest variance unbiased estimator can cause the Poisson PMF to more
steadily and accurately simulate the stochasticity of soil erosion events
over long-term observations than the binomial PMF.
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 λ=np in the Poisson PMF is an
unknown constant, the extension of monitoring period could lead the
numbers of times of I events (n) to approach infinity, and then the
probability (p) of R or S event generation would have to approach
zero. The above inference coincides with the assumption about the decreasing of
sediment generation in the closed-plot system, and further shows that the Poisson
PMF is more reliable to simulate the stochastic erosion events at
longer temporal scale.
The effect of influencing factors on erosion stochasticity
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 implication of integrated probabilistic assessment
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.
Conclusion
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.