Notice: Wiley Online Library will be unavailable on Saturday 27th February from 09:00-14:00 GMT / 04:00-09:00 EST / 17:00-22:00 SGT for essential maintenance. Apologies for the inconvenience.
 Threshold crossings are crucial in many geophysical processes, including soil water balance, vegetation stress, landslides, particle erosion and transport. We focus here on the crossing properties of systems with additive or multiplicative stochastic impulses having the form of a marked Poisson process. We derive exact expressions for the probability of excursions of very short duration above/below a generic threshold. These expressions, together with mean first passage times, allow for approximating the full probability density function of excursion times with two-parameter distributions. Such approximations are effective for a wide range of stochastic processes covering relevant geophysical applications, as confirmed by the two examples provided regarding the dynamics of soil moisture content and water table depth.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 An important aspect of stochastic dynamics lies in the properties of threshold crossings. These properties are especially relevant to the study of geophysical processes having, or being influences by, threshold behaviors as discussed by Zehe and Sivapalan . Crossing properties are often characterized by the probability distribution of the time required to cross a threshold starting from a generic initial point. Noteworthy results concerned with the moments of such distribution include, e.g., the mean first passage time for non-Markovian processes [Masoliver, 1987] and for Poisson processes with exponential jumps [Laio et al., 2001b]. McGrath et al.  derived all the moments of the probability distribution of first upcrossing times, with specific reference to the case of linear systems driven by Poisson noise. All these works are related to dynamical systems with additive disturbances.
 We study here the excursion times, intended as the time intervals spent above/below a threshold, of dynamical systems forced by a marked Poisson noise, either additive or multiplicative. We derive exact expressions for the probability of excursions of infinitesimal duration; such expressions, together with mean first passage times, enable the approximation of the probability density function of excursion times with parametric distributions having two parameters. These functions, although not providing a conclusive solution to the complex problem of crossing properties of Poisson-driven systems, are sound approximations based on the dynamical systems characteristics and represent a significant improvement compared to the standard assumption of exponentially distributed excursion times [see, e.g., Botter et al., 2007]. Examples of ecohydrological processes that can benefit for the solutions proposed here include surface runoff generation, water soil infiltration and leaching, vegetation water stress and anoxic stress, and hydroperiods (or time intervals spent above the soil surface by a fluctuating water table).
2. Dynamical System, Crossing Properties and Distribution of Excursion Times
 Consider the generic dynamical system whose evolution is described by
where x is the state variable, f(x) is the loss function, g(x) is either a constant or a state-dependent function, and (t) is the stochastic forcing, having the form of a marked Poisson process, with constant rate λ and exponentially distributed depths with mean α.
 Considering dynamical systems having a stationary probability distribution, under the Stratonovich interpretation of the noise term the long-term probability density function (pdf) of the variable x reads [Van Den Broeck, 1983]
where C is the normalizing factor for the pdf to have unitary area, and the corresponding cumulative distribution function is PX(x) = p(u)du.
 When the function g(x) is positive-valued, the process in equation (1) can be simplified to avoid the multiplicative form of the noise. Substituting 1/g(u)du = z and rearranging terms, the original process can be rewritten as
where the Poisson noise (t) appears now in additive form. The resulting dynamics of z is different than the one of x in terms of magnitude of the disturbances, but not in their time of occurrence.
 The mean frequency of upcrossings (or downcrossings) of a generic threshold, ν(ξ), equals the long-term pdf times the loss function of the (equivalent) additive system, both evaluated at the threshold, ξ [e.g., Laio et al., 2001b], ν(ξ) = p(ξ)f(ξ)/g(ξ). In general, the time interval elapsed between an upcrossing and a subsequent downcrossing of a threshold ξ is a stochastic variable, τa, whose average value is the mean first passage time (MFPT) across the threshold ξ from above, Ta(ξ); this average reads [Laio et al., 2001b]
Similarly, the time interval elapsed between a downcrossing and a subsequent upcrossing is a stochastic variable τb whose average value is
 Both stochastic variables, τa and τb, are positive-valued and have a non-null probability associated to their left limit value, i.e., the infinitesimal excursion time, which can be derived from the basic properties of the dynamical system. The probability of having a downcrossing of ξ immediately followed by an upcrossing depends on the properties of Poisson noise but not on the state of the system; the probability density for τb = 0 simply reads
because any jump will produce an upcrossing when x is just below ξ.
 Conversely, the probability of having an upcrossing of ξ immediately followed by a downcrossing is related to the loss function of the system and it can be derived from the distribution of noise jumps. Let us consider a jump across the threshold, bringing the system at an infinitesimal (positive) distance dhτ from ξ; the following downcrossing occurs after an infinitesimal time dτ, with the ratio of the two, dhτ/dτ, being equal to the loss function of the dynamical system (see Figure S1 in the auxiliary material). When the noise is additive, the probability distribution of the jumps above threshold is exponential, due to the properties of the Poisson noise, and an infinitesimal jump has a probability density of 1/α. The probability density of an infinitesimal excursion time above the threshold ξ thus reads = f(ξ)/α. When the noise is multiplicative, the distribution of jumps above threshold is not exponential due to the distortion in magnitude imposed by the multiplicative factor. However, if one refers to the equivalent dynamical system (3), where time scales are invariant, the probability density of an excursion of infinitesimal duration above the threshold ξ reads
which includes the case of additive noise, with g(·) taking a unitary value. The same result can be obtained analytically as shown in the auxiliary material (section S1).
 The probability density for τ = 0 is a key property for characterizing the distribution of excursion times: = 1/Ta,b(ξ) identifies exponential distributions, associated to memoryless processes, while greater (smaller) values identify super-exponential (sub-exponential) behaviors, where probabilities higher than 1/Ta,b(ξ) can be a proxy for clustering [Kagan and Jackson, 1991].
 Knowing the mean excursion time above/below a threshold and the probability of an infinitesimal duration, one can approximate the full probability density function of excursion times with a two-parameters distribution. A commonly used parametric distribution having a non-null and finite value in the origin, is the double exponential function with even weights, i.e.,
 The two parameters θ1 and θ2 can be derived equating the mean value of pτ(t) with the MFPT given by equations (4)–(5) and the initial value, pτ(0), with equations (7)–(6); the resulting parameters of the distribution (8) thus read
This expression is valid for · Ta,b(ξ) ≥ 1, thus for super-exponential behaviors; this is always satisfied for excursions below threshold, since necessarily Tb(ξ) > 1/λ, but not always for excursions above threshold. When the condition is not satisfied, the distribution of excursion times might be non-monotonic and it cannot be modeled appropriately by equation (8). An alternative function is the generalized Pareto distribution, with parameters defined by imposing the mean value and the initial value of the pdf; the function reads
 Fitting capabilities of the double-exponential and Pareto distributions are compared in the next section with the simple exponential distribution pτ(t) = 1/Ta,b(ξ) · exp[−t/Ta,b(ξ)], which is routinely used in standard applications [e.g., Botter et al., 2007].
3. Applications and Discussion
 The first application considers the soil water balance at a point, whose excursion times are relevant, e.g., for the investigation of surface runoff generation, water soil infiltration and leaching, and vegetation water stress. The soil moisture dynamics read
where s(t) is the relative soil moisture, n is the soil porosity, Zr is the active soil depth, E(s) is the evapotranspiration rate, L(s) is the leakage from the active soil, and ψ(t) is the net rate of infiltration from rainfall, having the form of a marked Poisson process with constant rate λ and exponentially distributed infiltration events with mean α. The model is reported in the auxiliary material (section S2) and detailed by Laio et al. [2001a].
 Numerical simulations of the soil moisture dynamics are performed in order to evaluate the properties of excursion times for the system in equation (1) with the substitutions
In Figures 1a and 1b the simulated mean excursion times above/below different thresholds correctly superimpose to the expressions from equations (4)–(5) and also the simulated probabilities of infinitesimal excursion times are in very good agreement with equations (6)–(7). The analytical properties are then used to parameterize the double exponential function (8), the exponential and the Pareto distributions (10). Figure 1 compares the numerical and analytical standard deviations (std) to verify the fitting capability of the approximations proposed: results confirm the goodness-of-fit of the double exponential function, when the condition · Ta(ξ) ≥ 1 is satisfied, while in some cases the Pareto distribution has a diverging variance.
 Normalized frequency histograms of simulated time intervals and probability density functions are compared for a sample soil moisture threshold in Figures 2a and 2b which show a very good agreement between simulations and the proposed distributions. In some cases (Figure 2b) also the exponential function fits well the simulations: this occurs when the MFPT is small, i.e., few days (compare with values in Figure 1b). For such fast dynamics and short time intervals, the deterministic dynamics of the system becomes less relevant and the crossing properties are mainly regulated by the Poisson noise, with exponential interarrival times. In cases with larger MFPTs (Figure 2a), the exponential distribution is completely inadequate to describe the distribution of excursion times. The double exponential and the Pareto distributions, having two calibration parameters, are more flexible and always behave better than the single exponential.
 The differences between the double exponential and the Pareto distribution can be better studied focusing on the tails of the distributions, at large excursion times. For this purpose, the cumulative distribution functions are compared with the empirical distribution function, Pi = i/(n + 1), where i is the position of each value in the sorted sample and n is the sample size. Cumulative distributions are plotted on a probability chart having on the vertical axis the quantity −log(1 − P) which highlights large excursion times. Insets in Figures 2a and 2b show an example of such comparison for the soil moisture dynamics where the exponential nature of the tails is clear. Such nature cannot be captured by a power-law distribution such as the Pareto, and the double exponential function overperforms the single exponential thanks to the higher flexibility given by the second parameter.
 A second application considers the case of a system with a state-dependent function g(x) multiplying the noise: a water balance model involving the dynamics of a fluctuating water table. In this case, excursion times are crucial to characterize hydroperiods (intervals of time during which water stands above ground) which in turn control plant anoxic stress and the spatial structure of vegetation in wetlands [Todd et al., 2010]. The water balance equation reads
where y is the water table depth (intended as the separation between saturated and unsaturated soil), Re is the stochastic groundwater recharge (marked Poisson process with constant rate λ and exponentially distributed events with mean α), ET is evapotranspiration, fl is the lateral flow to/from an external water body, and β is the specific yield, which modulates the change of water table depth in relation to the volumetric change of water in the soil column. The system is described in the auxiliary material (section S3) and studied in detail by Tamea et al. .
 Crossing properties are now calculated using
MFPTs (equations (4)–(5)) and the probability of infinitesimal excursion times (equations (7)–(6)), are used again to fit the double exponential probability distribution, the exponential and Pareto distributions. Figures 1c and 1d show the superimpositions of analytical and numerical properties, while the comparison between numerical and approximated standard deviations confirms the capability of the double exponential function to describe higher order moments of the distribution of excursion times, while the Pareto distribution has a diverging variance in most cases of excursions above threshold.
Figures 2c and 2d show the comparison between the normalized frequency histograms of simulations and the probability distribution functions for the water table dynamics across the ξy = 0 threshold, identifying soil surface saturation and hydroperiods. The simple exponential function is inadequate to describe the distribution of time intervals in both cases, while the fitting capabilities of the double exponential function and the Pareto distribution are equivalent at short time scales. The cumulative distributions in the insets show that also in the water table case, right tails of the simulation have exponential behavior, and such nature can be captured only by the double exponential function.
 In conclusion, the MFPT may not be sufficient to study the crossing properties of geophysical systems forced by Poisson noise. An additional property has been described here, which is the probability associated to excursions of infinitesimal duration above/below threshold for both additive and multiplicative noise. This property enables the approximation of the full probability distribution of excursion times using suitable two-parameter functions, such as the double exponential or the Pareto distributions.
 These functions generally model well the probabilities of short excursion times (comparable with the magnitude of the MFPT) providing an effective approximation at time scales which are usually of highest interest. At longer excursion times, the goodness-of-fit varies across different thresholds and model parameters, but the exponential nature of the right tails is captured only by the double exponential distribution.
 Given the intrinsic complexity of the problem, the approximations proposed are likely to have extended use in many geophysical applications, although additional research is needed to provide a better characterization of the right tail of the excursion distribution.
 Authors gratefully acknowledge two anonymous reviewers for their thoughtful comments which helped to improve the paper.
 The Editor thanks the anonymous reviewers for their assistance in evaluating this paper.