An observation-based stochastic model for sediment and vegetation dynamics in the floodplain of an Alpine braided river

Authors


Abstract

[1] Riparian vegetation dynamics in Alpine rivers are to a large extent driven by the timing and magnitude of floods which inundate the floodplain, transport sediment, erode the river bed, and create and destroy suitable germination sites. Here we present a stochastic approach for studying sediment-vegetation dynamics lumped at the floodplain scale and driven by stochastic flood disturbances. The premise of the model is that floods erode riparian vegetation in the inundated part of the floodplain and expose bare sediment surfaces. In the absence of subsequent flooding these surfaces are gradually recolonized. The stochastic nature of the disturbance process and the deterministic rate of vegetation colonization are described by a Poisson arrival of floods and a process equation which treats the floodplain erosion and vegetation colonization processes, respectively. An analytical solution is developed to obtain the probability density function of the exposed sediment area. The model is applied to the Maggia River in Switzerland, where it reproduces the changes in riparian vegetation cover observed from aerial photographs with an absolute error less than 5%. The model has potential as a tool to study the impacts of changes in the disturbance regime on sediment and vegetation dynamics.

1. Introduction

[2] River flow and alluvial sediment dynamics in Alpine braided rivers are key for sustaining the ecological biodiversity of the riparian corridor [e.g., Poff et al., 1997; Friedman and Lee, 2002; van der Nat et al., 2002]. In climates where water availability is not the limiting factor for growth, it is the erosion of plants and their sites by floods, as well as the creation of new sedimentary deposits that dominates vegetation recruitment [e.g., Polzin and Rood, 2006]. Floods play a fundamental role in this process by eroding the channel bed and vegetated islands and bars, by creating suitable colonization sites on new deposits, and by determining the time available for colonization of those sites into more or less stable patterns through the interarrival time between floods [e.g., Resh et al., 1988; Bendix and Hupp, 2000; Friedman and Lee, 2002; Nilsson and Svedmark, 2002]. Moreover, the interarrival time between floods contributes to modulate the density and structure in vegetation composition on gravel bars, because the rate of flood recession influences the available soil moisture and, in turn seed germination [Mahoney and Rood, 1998].

[3] The result of these complex interactions is a mosaic of vegetation patches on the floodplain with different species tolerance to disturbance by inundation or erosion. Surfaces closest to the river are most frequently inundated and usually contain pioneer herbaceous plants and shrubs (e.g., Salicaceae family), those at higher floodplain elevations are inundated less frequently and are covered by shrubs and flooding tolerant riparian trees [Auble et al., 1994; Poole et al., 2002; Glenz et al., 2006]. These features classify braided gravel bed rivers among the most dynamic fluvial environments from the point of view of geomorphological, hydrological and ecological diversity. At the same time they are among the most endangered ecosystems, threatened by flow regulation, habitat alteration and species invasion [Tockner and Stanford, 2002].

[4] Many Alpine rivers have been subject to water withdrawals and regulation for different purposes (hydropower, urban use, agriculture, etc), which have heavily affected natural hydrologic regimes [Poff et al., 1997]. The impacts of these changes on the riparian ecosystem are difficult to quantify because of the inherent complexity of riparian vegetation dynamics and of the numerous interactions and feedbacks among different processes (e.g., erosion, sedimentation, vegetation colonization and growth, uprooting, etc.). Often river regulation is associated with a reduction of flood magnitude and frequency and results in a gradual colonization of the floodplain because the role of colonization by vegetation prevails over that of disturbances that rework the river morphology [e.g., Poff et al., 1997; Shafroth et al., 2002; Molnar et al., 2008]. A similar effect has also been recently reproduced in flume experiments [Gran and Paola, 2001; Tal and Paola, 2007] although at this scale the vegetation root growth rate may play a critical role. The study and modeling of such processes is therefore important for both speculative and applicative reasons (e.g., river restoration).

[5] Many hydroecological models have been developed in recent years in order to investigate vegetation and river dynamics [e.g., see Glenz, 2005; Camporeale and Ridolfi, 2006; Muneepeerakul et al., 2007a, 2007b]. Detailed semideterministic models aim at a spatially explicit description of the above interactions [e.g., Glenz, 2005], but usually require heavy parameterization [Dixon and Turner, 2006]. In this case calibration can be seriously compromised by the often limited amount of available data. Neutral metapopulation models have proven quite successful in explaining the biodiversity that is observed at the catchment scale [Muneepeerakul et al., 2007a, 2007b]. Stochastic approaches based on dichotomic noise processes [e.g., Horsthemke and Lefever, 1984] have also been proposed to explain the variability of the riparian corridor at the transect scale and across river cross sections [Camporeale and Ridolfi, 2006]. Such a model was also extended to the floodplain reach scales [Muneepeerakul et al., 2007c]. An alternative approach is to focus at the river reach scale by coupling the disturbance regime with simple succession rules for the prevalent vegetation species. A recent example of this approach is the structured population model of Lytle and Merritt [2004] to simulate the annual abundance of cottonwood.

[6] A similar minimalist approach is taken in this paper in order to investigate the effects that water regulation has on geomorphic processes [e.g., Merritt and Cooper, 2002]. Specifically, we aim at quantifying changes that have been observed from historical records of aerial photographs. We adopt as an exemplary case the Maggia River (Canton Tessin, Switzerland, Figure 1) where a complex hydropower system has been built in the headwaters of the basin and where a long record (1933–2004) of aerial photographs is available. In the valley, the postdam flow regime has clearly affected the dynamics of sediment transport, river morphology, groundwater recharge, and fluvial ecotone biodiversity [Molnar et al., 2008; Ruf et al., 2008]. Moreover, floods have been shown to significantly impact the ecology of the riparian corridor [Bayard and Schweingruber, 1991]. This motivates the need to formulate a tool able to estimate the area of exposed water and sediment (or its complement, i.e., the area covered by vegetation) in relation to the statistics of the river flow regime. We think that a simple model with these characteristics would represent a powerful tool for negotiating the renewal of concessions for the water use in the valley.

Figure 1.

(left) Location of the Maggia valley in Switzerland with the braided study reach around Someo (46°17′14″N, 8°39′44″E). (top right) Typical annual hydrograph for the predam period, (middle right) typical annual hydrograph for the postdam period showing the disappearance of the seasonal component, and (bottom right) daily mean annual hydrograph behavior for both predam (dashed line) and postdam (solid line) periods; Tw and Tc denote the warm season and the cold season, respectively.

[7] In this paper we develop a stochastic model to describe the spatially lumped dynamics of exposed sediment and water in the braided reach of the Maggia River in relation to the history of river flow disturbances. The model uses a flow vs inundated area relationship at the site to give physical meaning to the inundation process. The resulting model is parsimonious in the parameters, which is a key features that allows for reliable calibration even when only few observations are available. Moreover, this modeling approach allows for a fully theoretical formulation, which may be solved analytically for statistical equilibrium conditions. We thus arrive at the probability density function of the exposed sediment area as a function of the statistics of the disturbance regime and the vegetation colonization rate parameter.

[8] Despite simplifying the real process complexity, such a model provides insights into the amount of stochasticity that can be expected in the erosional dynamics of vegetated alluvial noncohesive sediment typical of Alpine braided rivers [van der Nat et al., 2002]. As such, this approach represents a first step toward the development of more realistic models e.g., accounting for the role of the mechanical anchoring of vegetation roots.

2. Observation-Based Model Philosophy

2.1. Study Area and Data

[9] The Maggia valley (Canton Tessin, Switzerland, Figure 1) has one of the last natural (not urbanized) floodplain forest and riparian ecosystem in Switzerland, and has been declared a site of national importance [Kuhn and Amiet, 1988]. The basin of the Maggia covers an area of approximately 930 km2. The Maggia river flows for about 50 km from the Naret lakes (2240 m a.s.l.) down to Lake Maggiore (193 m a.s.l.), and has three main tributaries (Lavizzara, Bavona and Rovana). Since 1953, the streamflow regime in the valley has been modified by the development of a complex hydropower system in the headwaters [Molnar et al., 2008]. The strong seasonal snowmelt component has been reduced to a rather constant base flow release with occasional flood peaks (see the right plots in Figure 1). Later changes of the hydrological regime are due to the introduction of the environmental (minimum) flow requirements (1969, hereafter referred to as EFR) and some refinements that have led to the present releases of 1.8 m3 s−1 for the summer period and 1.2 m3 s−1 for the rest of the year [Pfamatter and Zanetta, 2003]. Flow regulation by dams has altered the temporal correlation of streamflow, and enforcement of the EFR tends to fix the position of the groundwater table. This allows vegetation growth to be practically decoupled from streamflow and to depend mostly on groundwater and water recharge from lateral slopes. Most of the tributaries from the side valleys are indeed either diverted into the hydropower water distribution system, or their discharge infiltrates completely into the coarse alluvial aquifer of the main valley before reaching the Maggia River in dry periods. Under flood conditions, almost all water is released into the main river channel.

[10] We investigate the floodplain reach between Riveo and Giumaglio, which is one of the most dynamic parts of the Maggia floodplain (Figures 1 and 2a). Here the river is braided and characterized by a relatively coarse, rounded and uniform noncohesive sediment with a mean diameter of 120 mm. This section is 2.8 km long and covers a total surface Af = 1.58 km2. Two-dimensional hydrodynamic simulations (Figure 2b) performed by the finite volume shallow water flow model 2dmb (new version is BASEMENT, http://www.basement.ethz.ch/) have shown that the whole area is inundated by a flood of magnitude 1000 m3 s−1 [Ruf et al., 2008].

Figure 2.

(a) Domain of analysis and related division into subdomains (I–IV). Blue shows the portion of water and sediment that did not change between 1995 and 1999; green shows the area that was water and sediment in 1995 and has been colonized by vegetation between 1995 and 1999. (b) Rating curve relating the total area A(q) inundated to the corresponding discharge q (from 2-D hydrodynamic modeling) and related power law approximation.

[11] Sagebrush (Artemisia campestris, Asteraceae) and rosemary willow (Salix elaeagnos, Salicaceae) together with blackberry bushes are the most common species to take root in the coarse gravel soils of the studied domain. As some native grass species, this vegetation can germinate and grow even along shorelines. Alluvial terraces are also occupied by Salix elaeagnos, Alnus incana and Populus nigra, all trees with a rapid growth rate. High and rarely flooded terraces show mature soil on which ash (Fraxinux excelsior), Oak (Quercus robur), small-leaved lime (Tilia cordata) and Scots pine (Pinus sylvestris) are the more commonly found species. The investigated domain is largely undisturbed, the only anthropogenic influence being the presence of three embankments, whose influence on the inundation dynamics by floods is however considered in the hydraulics rating curve (Figure 2b).

[12] Ten aerial photos (1933, 1944, 1962, 1977, 1978, 1989, 1995, 1999, 2001, 2004) and a detailed manual field survey (performed in 2006) are our main database of the observed morphological changes. Of these, nine observations (1962–2006) belong to the postdam period. We georeferenced the available aerial photographs. For the georeferencing, we first identified a number of control points in the orthophoto Swissimage (Swisstopo) which has a ground resolution of 0.5 m and standard deviation for the precision in position in flat terrain of ±0.25 m. The control points were located on easily identifiable fixed landmarks (roads, houses, bridges, etc.) in the floodplain area only. We georeferenced all vegetation grids from aerial photos to match these control points with a second-order polynomial transformation in ArcGIS. We evaluated the position accuracy by selecting at random additional visible points in the maps and computing the root mean square location error, which was approximately 5 m, with a standard deviation of 3 m. Since the smallest mapped vegetation polygons have length scales on the order of 102 m, we consider that statistically the position error does not play a big role in the results.

[13] On the georeferenced aerial photographs we identified four surface classes (water and sediment, grass, shrubs and forest) by a combination of automatic image processing and manual verification. From these maps we then detected and quantified changes in vegetation cover and the rate at which exposed sediment areas are recolonized (e.g., see Figure 2a). We recognized that these changes mirror also changes in the flow regime (see, for example Figures 3a–3b) [Molnar et al., 2008]. However, the presence of vegetation species and the geomorphic changes of the related sites do not seem to show strong dependence in the downstream direction. The persistence (here intended as the tendency of a given class to not change location among years) of shrubs in the domain (Figure 3c) clearly shows where stable islands have formed without the appearance of evident longitudinal patterns. We profit of this conclusion, in order to virtually increase the number of observations. That is, we split the whole domain into four morphologically similar parts (Figure 2a) and consider the evolution of each of them as being independent of the others.

Figure 3.

Evolution of the submerged and exposed sediment areas for a reach of the river Maggia (a) before and (b) after a major flood event. (c) Map of shrub persistence given by the frequency each point was classified as covered by shrubs in the period 1962–2004 (from aerial photographs). The legend refers to the number of times the same pixel was occupied by the same class in successive photographs.

2.2. Modeling Hypotheses

[14] Consider the floodplain reach shown in Figure 2a whose area Af is potentially inundated and reworked by some extreme flood event. As an example Figures 3a and 3b show the consequences of a large flood recorded in 1978. Figure 3a (taken on 27 June 1977, daily average streamflow Q ∼ 16 m3 s−1) shows (in blue) the portion of Af made up of sediment and water Asw, whereas the complementary area (in beige) Av is occupied by classes of vegetation. Thus, Af = Asw + Av, or by normalizing with respect to the floodplain available area 1 = ωsw + ωv, with ωsw = Asw/Af and ωv = Av/Af. A successive flood event (7 August 1978, Qpeak = 673 m3/s for a precipitation event of 500 years return period) renewed the floodplain morphology and exposed new sediment as a consequence of both erosion and sedimentation. The increase in the area Asw is visible in Figure 3b (taken on 12 August 1978, daily average streamflow Q ∼ 13 m3 s−1), which corresponds better to the area Amax inundated by the maximum daily mean flow Qmax rather than to the area Ap inundated by the peak flow Qp. For instance, for changes that occurred in another flood (1989) the ratio between the actual exposed area Asw and Amax is 92.5%, whereas the ratio Asw/Ap decreases to 82%. These values depend on the ratio Qmax/Qp which varies in the range 0.6 ÷ 0.9 for the available data. Thus, while the area Ap inundated by Qp generally would overestimate the actual reworked portion of the floodplain, the area Amax inundated by Qmax accounts for the inherent complexity linking erosional processes to their causal floods.

[15] We model the evolution of the exposed area of sediment and water Asw in the domain Af resulting from the coupled action of maximum daily mean flood disturbances Qmax, and the tendency of riparian vegetation to recolonize the new exposed sediment. Because observations have shown these processes to be dependent on climate, site and hydrologic regime [Shafroth et al., 2002; Friedman and Lee, 2002; Johnson, 1994; Friedman and Auble, 1999; Pizzuto, 1994], the following working hypotheses are made.

[16] 1. Years are divided into a cold season of duration Tc (i.e., December–March) and a warm season Tw (April–November). Interactions between vegetation, floods and sediment occur only during the warm season (during the cold season vegetation is dormant and flow measurements show that floods are practically absent; e.g., see Figure 1). For the sake of modeling convenience, we assume vegetation being active until November.

[17] 2. The alluvial material has negligible cohesion. Only flood events exceeding a given threshold flow rate Q* are capable of potentially exposing new sediment through erosion and sedimentation.

[18] 3. Flood disturbances occur as a stochastic process with uncorrelated magnitude and interarrival times. This hypothesis holds very easily for the postdam period.

[19] 4. Colonization by vegetation occurs deterministically in between flood events given the type of vegetation species that are present in the valley and the related climate.

2.3. Stochastic Flood Disturbances and Deterministic Colonization

[20] We consider a time sequence of daily mean streamflow {Q}(t) in the postdam period. Because of the flow regulation, the runoff regime shows flashy features, with occasional high flows. We extract a discrete process {q}(ti) from the streamflow Q(t) as

equation image

where τ(Q) is the total duration of the event for which Q > q*, and q* is a threshold whose physical meaning is ascribable to incipient bed load transport conditions. Although we examined the threshold q* in terms of incipient motion conditions and found an approximative value of q* = 50 m3/s, such a result can be considered as indicative only. Hence, in the following we will treat q* as a model parameter and obtain its numerical value either by model calibration or by direct estimation from aerial photographs (section 4).

[21] For a given event, only the maximum daily flow occurring at the time t = ti is considered to define the stochastic process q(ti) (e.g., see Figure 4). We obtain the statistical properties for both the magnitude and the interarrival time between the events (Figure 5). The sample autocorrelation function of streamflows (not shown) suggests that events above 40 m3/s are practically uncorrelated.

Figure 4.

Example of the discretization of the daily mean river hydrograph for q* = 40 m3 s−1. This value of q* is only indicative here.

Figure 5.

Histogram for (a and b) the distribution of flood event magnitude and (c and d) interarrival time for the Maggia River postdam period and related exponential approximation. These functions refer to thresholds q* = 76 m3 s−1 (Figures 5a and 5c) and q* = 175 m3 s−1 (Figures 5b and 5d) as obtained from model calibration and direct estimation from aerial photographs (section 4).

[22] In order to proceed analytically, we approximate the interarrival time between the disturbances with an exponential distribution being aware that this assumption would become critical for thresholds lower than 40 m3/s. The exponential distribution allows to consider q(t) occurring with a probability ρΔt, that is a stochastic Poisson process of rate ρ [e.g., Cox and Miller, 1965]. This process is marked by the event magnitude distribution b(q), also exponential in this case

equation image

The pdf b(q) has mean μq = equation image and variance σ2q = equation image, and has the advantage of having only one fitting parameter, λ.

[23] From 2-D hydrodynamic modeling, the flow magnitude q is now associated with a “rating curve” relating the area inundated to the corresponding discharge A = A(q) (Figure 2b). Notice, that the area A(q) does not necessarily correspond to the area covered by water and exposed sediment (Asw). The function A(q) has been investigated under several contexts, e.g., hydraulics, geomorphological and ecological [van der Nat et al., 2002; Ruf et al., 2008]. For braided rivers this relationship is reasonably approximated by the power law A = k1qk2, with k2 < 1. In our case, A(q) has been determined for the whole domain (Figure 2b) and for each subdomain by 2-D hydrodynamic modeling [e.g., Ruf et al., 2008] in order to estimate the parameters k1 and k2 of the related power law approximations. Because of channel reworking during floods, the function A(q) can change for low values of q and for individual short river sections. However, for sufficiently high q the A(q) relationship is mostly determined by the balance between the mean river bed slope and the required channel conveyance capacity and it can be considered nearly independent of the local morphology.

[24] The pdf of the area inundated by flood events g(A) can readily be obtained from that of floods by means of the function A(q). To this regard, we redefine the function A(q) as both lower and upper bounded:

equation image

i.e., at A = A* and at A = Af, respectively. The first limit is due to the threshold q*, while the second one becomes effective when the whole floodplain is flooded by a corresponding event of magnitude qqf. The pdf of A can thus be obtained by applying the derived distribution approach to (2) and (3), and imposing the probability boundaries Pr[AAf] = 1 and Pr[AA*] = 0. This leads to a pdf for A made up of a continuous part gc(A) and an atom of finite probability gat(A) [e.g., Cox and Miller, 1965] at the upper boundary Af:

equation image

whose coefficients a0, a1, a2 are reported in Appendices A and B.

[25] The continuous part is

equation image

valid in A ∈ [A*, Af), whereas the probability gat(A) of flooding the whole area is

equation image

Notice that g(A) satisfies the condition equation imageg(A)dA = 1. The cumulate of g(A) is the probability distribution function

equation image

[26] We idealize the colonization of the floodplain by vegetation to take place deterministically and continuously between disturbances. The literature offers several options to describe this process including the JABOWA forest growth equation [Botkin et al., 1972]. This equation can be approximated by the Verhulst-logistic equation [Hunt, 1982] or by its generalized form [Camporeale and Ridolfi, 2006]. Common to such models is the description of a growth mechanism that is a function of the ability of the given species to tap soil moisture and of the carrying capacity, that is, the maximum sustainable biomass supported by the local environment. Thus, the rate of growth of vegetation initially increases for low density of vegetation coverage (i.e., water and nutrient resources are not limiting), reaches a maximum and then decreases as the carrying capacity of the local environment is approached.

[27] Our system has open boundaries where germination may occur by several agents (e.g., wind, flow, woody debris). The resource-limited condition is therefore not particularly representative for such an environment. Also the magnitude of historical floods was never sufficient to completely remove vegetation from the entire domain. Hence, in our lumped approach the spreading dynamics of (low) vegetation species (e.g., grass and shrubs and young trees) that are active over the whole domain is summarized by a unique parameter. That is, we assume the colonization rate to be a function of the exposed area Asw only through the parameter k, i.e.,

equation image

This simple exponential colonization model however accounts for the saturation effect of the growing biomass as it colonizes the available exposed surface Asw.

2.4. Process Equation

[28] We write the equation of the process by considering the way the exposed area Asw changes in time as a result of both the stochastic and the deterministic actions, i.e.,

equation image

The rate of change of Asw by colonization by vegetation is then expressed by equation (9), whereas the rate of change of Asw as a result of flooding events is

equation image

The total rate of change of Asw is then

equation image

together with the boundary condition Asw(t) ≤ Af for any time t.

[29] The suffix i represents the occurrence of a disturbance with probability ρdt. The Heaviside function Θ(·) ensures that new sediment is exposed only if the effects of flood qi are such that the area exposed by it exceeds the current value; that is, only if A(qi) > Asw(ti). This feature makes our process structurally different from others [Cox and Miller, 1965; Rodriguez-Iturbe et al., 1999; Perona et al., 2007].

[30] Figure 6 shows an example of the numerically generated process in the period Tw showing clearly that the probability density function of the area A inundated by an event differs from that of Asw, because the latter is continuously subjected to recolonization. The process described by equation (11) shows statistical stationarity (i.e., no trends occur over the long term) because of the boundaries. Stationarity holds for any value of the constant k in the range [0, +∞), with a completely deterministic asymptotic dynamics for k = 0 or k → ∞. In such cases the pdf of Asw degenerates into the atom of finite probability of being at the upper or at the lower boundary, respectively. Between such two limiting cases the model gives rise to an interesting stochastic dynamics characterized by an atom of finite probability for Asw of being at Af, and a continuous pdf of having both vegetated and exposed sediment areas on the floodplain.

Figure 6.

In the process not all the events (dashed lines) are successful in increasing the exposed sediment area, but their effect is conditional to the actual exposed area Asw. The pdf on the left-hand side is that of “jumps” A(t), whereas the one on the right-hand side is that of the resulting process Asw(t).

3. Probabilistic Formulation and Steady State Solution

[31] The process for Asw has a pdf composed of a continuous part pc(Asw) and of two atoms of finite probability pat of being at zero and at Asw = Af:

equation image

The atom of probability at Af only depends on the statistics of floods inundating an area bigger than Af, and is given by

equation image

[32] We compute a local balance of probability for the distribution of Asw(t) and obtain the master equation of the process [e.g, Cox and Miller, 1965; Rodriguez-Iturbe et al., 1999], i.e., the following partial integral differential equation for the continuous part of the pdf of Asw,

equation image

and

equation image

as far as the atom of probability at zero is concerned. The variable a in equation (14) is the dummy variable of integration.

[33] Because of the exponential type of decay imposed by equation (9), the boundary at zero is only asymptotically approached. Moreover, f(0) = 0, and the atom of probability decays in time as

equation image

which shows that this contribution either is always zero or will exponentially vanish in time if the process starts at zero.

[34] We look for the asymptotic solution when t → ∞ and starting with the process away from zero (e.g., p0at(0) = 0). The seasonal-dependent process dynamics would require to account for the statistical initial condition at the beginning of each season. This can be done by computing a local balance of probability between the pdf's at the end of the cold and the warm seasons [e.g., see Perona et al., 2007]. However, since in this case the whole process is inactive during Tc, the pdf at the end of the cold season is the same as that at the end of the warm one, and the matching condition is a simple time renormalization. Hence, statistical stationarity can be studied by ignoring the cold season.

[35] By substituting f(Asw) with −kAsw and using the probability distribution function (5) for gc(A), equation (14) simplifies to the following integrodifferential form:

equation image
equation image

valid for Asw ∈ [0, Af). In order to solve this equation we first of all consider that the cumulate of the unknown pdf pc(Asw) is

equation image

Then, we integrate the equation to obtain

equation image

[36] Taking advantage of the integration per part rule and of equation (18), using p(Asw) = P′(Asw) and doing some algebra we obtain the following first-order ordinary differential equation with variable coefficients for Pc,

equation image

valid for Asw ∈ [0, Af). By rewriting the equation and requiring that Pc(Asw) = 0 as Asw → 0 one sees that Const = 0, and the equation to solve is

equation image

together with the boundary condition Pc(Asw) = 1 − Θ(AAf)pAfat.

[37] The analytical solution for the CDF of Asw involves the exponential integral function Ei[·] [e.g., see Abramowitz and Stegun, 1965], i.e.,

equation image

The coefficients a2, a3, a4 and a5 of the CDF are shown in Appendices A and B, where as patAf is given by equation (13). The pdf of Asw can readily be obtained by computing the derivative of the CDF with respect to Asw and is therefore not shown here.

4. Results and Discussion

4.1. Model Validation

[38] We apply the model to simulate the dynamics of the interactions between flooding, erosion and colonization by vegetation of the study area by using the historical daily streamflows recorded in Bignasco at the upstream end of the study reach. In order to perform a comparative analysis with which to discuss the results we use q* and k as model parameters. In one case we estimate their value from aerial observations; in the other case we obtain their value directly from model calibration. One can assess the colonization rate ke (where the index e stands for estimated) from a number of consecutive photos with no floods in between. In our case, given the limited number of pictures and the fact that the lapsed time between our photographs is rather large, we consider the 4-year period 1995–1999. Within this period, the maximum registered (daily mean) flood is about 175 m3 s−1. Thus, by fixing qe* = 175 m3 s−1, a colonization rate parameter ke = 0.000614 d−1 is estimated from the exponential decay of the active process that joins the two observations. Obviously, this value of ke describes an average dynamics, because the actual dynamics between moderate floods of intensity smaller than qe* would be excluded. However, this strategy allows us to simulate the process and validate the model using historical data without calibration. The result of this simulation is shown in (Figures 7a–7e). We find a good correspondence between the model simulations and the observations both at the scale of the whole domain (Figure 7a) and the subdomains (Figures 7b–7e). The corresponding error map is shown in Figure 8. In particular, the model reproduces quite well the 20-year trend observed in the period 1978–1999. However, the sharp increment in the exposed sediment between 1977 and 1978 is only partially captured. This can be the result of an underestimation of the colonization rate due to the chosen qe* value. In order to explore this, the rest of this section shows results of calibrating the model to obtain the optimal value of the colonization rate.

Figure 7.

Comparison between (a) the process obtained with the estimated parameters for the whole domain and (b–e) subdomains I–IV and (f) the process with calibrated ones for the whole domain. Black dots are the data obtained from observations. The data from 2004 and 2006 corresponding to the dotted frame in Figure 7f are used for model validation. The two dashed lines indicate the upper boundary for the process and the lower one for disturbances.

Figure 8.

(top) Error map of model performance when using historical daily streamflow data for different pairs of parameters q* and k. (bottom) Plots of the error map surface according to the cross section shown in the top plot. For the sake of coherency with calibration these plots are built on 1962–2001 data only.

[39] Another way to validate the model is to first calibrate the parameters on a number of observations and then validate it on successive ones. Hence, we calibrate the parameters qc* and kc (where the index c stands for calibrated) by minimizing the mean square error computed by comparing the model results for different choices of qc* and kc against the observed points in the period 1962–2001 and for each part of the domain. Accordingly, the last two observations (2004 and 2006) will only be used for validation. By using the observations coming from the subdomains I–IV (see Figure 2a), the error map is built over a number of 28 observations rather than only 7. The best fit is obtained for the pair qc* = 76 m3 s−1 and kc = 0.0017 d−1 as also shown in the error map contour plot in Figure 8. Although a best fit can be found, it is also evident that the error map function has a rather flat minimum. Thus, on the one hand the fact that a minimum does exist plays in favor of the modeling hypotheses. On the other hand the flatness of the region around the minimum reflects the limitations of this model which reduces the complexity of the phenomenon to only two adjustable parameters. Furthermore, the contour plot shows that for low values of k, the model seems to become insensitive to the threshold chosen to identify the disturbances. This has a precise physical meaning, that is for very low colonization rates the dynamics of the exposed sediment and water do not fluctuate very much. Thus once a disturbance exposes a fraction of the domain, successive smaller events are basically ineffective because vegetation takes a very long time to recolonize the available area (see also section 2).

[40] Compared to the estimated parameters, model calibration gives a higher value for the colonization rate parameter kc associated to a lower values of the threshold q*c. The calibrated parameters allow the model to produce a better fit than the estimated ones (compare also Figures 7a and 7f). Moreover, the model is now able to capture the extreme event registered in 1978 and the related validation on the 2004 and 2006 data is also satisfactory. The analytical probability distribution functions corresponding to the two sets of parameters are shown in Figure 9, and allow the calculation of the probability of finding a given exposed area in the whole domain for the present hydroclimatic and water regulation conditions. For instance, the mean exposed area for sediment and water calculated from the aerial pictures is ∼525000 m2, which corresponds to a probability of 75%(40%) of having a larger exposed area if one uses the model with estimated (calibrated) parameters. Obviously, such an information is relevant to understand the present ecological status of the riparian corridor, as well as possible changes induced by different hydrologic regimes.

Figure 9.

Probability distribution functions of G(A) (solid line) and P(Asw) (dashed line) for the process with (top) estimated and (bottom) calibrated parameters. The related pdf's g(A) and p(Asw) are shown in the inset frames.

4.2. Relevance of the Model to Changing Hydrologic Scenarios

[41] A particularly interesting use of the model is in applications which investigate the effects of changes in the streamflow regime on the statistics of the evolution of the floodplain [Auble et al., 1994]. Such changes can be due to either climatic changes or new release policies for hydropower and other water uses [e.g., Friedman et al., 1995; Collier et al., 1997]. This is an appealing possibility offered by this simple model, that we present here in terms of a contour plot of the mean of Asw as a function of both mean interarrival time τ = 1/ρ and disturbance magnitude μb. This plot is shown in Figure 10 for the stationary solution, but the process can alternatively be simulated numerically, thus accounting for transient effects. The calculated means using both estimated and calibrated parameters are indicated by the black dot in Figure 10. The model with calibrated parameters (Figure 10b) has a mean exposed area μAsw = 486000 m2 that is actually very close to the value calculated from the aerial photographs in the post dam period. A slightly higher value μAsw = 598000 m2 is obtained when using the set of estimated parameters, which again can be justified by the rather low estimated colonization rate discussed previously.

Figure 10.

Mean of the exposed area Asw as a function of the disturbance parameters: model with (a) estimated and (b) calibrated coefficients. The black dot shows the present floodplain conditions as predicted by the two models.

[42] The plots of Figure 10 represent an interesting decision support tool based on the natural flow regime concepts [Poff et al., 1997; Collier et al., 1997]. Deciding the frequency and magnitude of controlled releases can have important implications with regard to future negotiation of the renewal of concessions for the future use of the water resources in the valley. By knowing the desired disturbance regime, a range of probabilities for the amount of exposed area Asw in the domain of interest can readily be calculated by using the plots of Figure 10. Vice versa, a desired mean exposed area of water and sediment (according to ecological needs, for instance) can be obtained for a potentially infinite combination of parameter pairs, or by constraining the choice to only a specific pair if also the higher moments of the distribution are fixed. Accordingly, the present model can be used to decide strategies for restoring water impounded environments. Decisional aspects in this sense concern the control of the channels narrowing effect and the maintenance of the braiding index [van der Nat et al., 2002; Jang and Shimizu, 2006], which are fundamental characteristics to maintain the ecological health of the riverine corridor and the preservation of the related biodiversity [Allen and Flecker, 1993; Auble et al., 1994].

5. Conclusions

[43] A simple stochastic model for the lumped water and sediment dynamics as modulated by vegetation colonization in Alpine environments has been studied in this work. We formulated the model with the purpose of minimizing the number of parameters and allowing for noninvasive model calibration from historical aerial photographs in the presence of few observations.

[44] This modeling approach allowed for an analytical formulation from which the analytical solution for the statistically stationary dynamic was obtained. The model can be used for predictive purposes to estimate the probability of having a given area of exposed sediment and water on the basis of the statistical properties of streamflow. As such, it is a useful tool to evaluate the effects of hydrologic changes, of either climatic or anthropogenic origin.

[45] By comparing the modeling results against observations for the Maggia Valley we quantified the reliability of the assumptions and the simplifications made. The model estimates the statistics of the exposed sediment and water on the floodplain, despite the conceptually simple erosional and colonization dynamics on which it is based. Although appropriate and useful in its present form, the model can be further improved as follows. A first improvement will be making explicit the dynamics of the different classes of vegetation that coexist in the lumped domain. This would help explaining the role of flood disturbances in the succession from herbaceous vegetation to woodland [Toner and Keddy, 1997]. To this end, the authors are currently extending the present model to a so-called master-slave dynamical model where the “Slave” component is represented by the fraction of vegetation evolving on the floodplain in a deterministic fashion.

[46] Future improvements also address the erosion processes of vegetated alluvial sediment. This can be achieved by explicitly accounting for both the correlation in the streamflow, their duration and the mechanical anchoring of vegetation roots growing on gravel bars. This step however requires also a detailed temporal information about floodplain changes. In turn, this highlights the importance of supporting this approach with remote sensing–based observations such as aerial and terrestrial photography.

Appendix A:: Coefficients of the Derived Distribution for A(q)

[47] The coefficients of the derived distribution for A(q) are

equation image
equation image
equation image

The mean μA is given by

equation image

where Γ(·, ·) is the incomplete Gamma function, and c1 = image and c2 = image The last term in (A4) is the contribution of the atom at A = Af to the mean. The variance for g(A) is

equation image

with obvious correspondence of symbols.

Appendix B:: Coefficients of the CDF Solution of Asw

[48] The coefficients of the CDF solution of Asw are

equation image
equation image
equation image
equation image

Acknowledgments

[49] This work was financially supported by the Swiss Federal Office for Forestry and Landscape (BUWAL), grant R-03-02. Franziska Stössel and Matthias Stutzenegger are also acknowledged for the floodplain survey, early data preparation, and the implementation of the 2-D water surface model. We also wish to thank Jorge Ramirez, Marco Marani, and an anonymous reviewer for their useful comments and suggestions.

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