Past analytical studies of meander planform development have mostly focused on the complexity of the governing equations, i.e., hydrodynamics, and less so on the stream bank resistance to erosion, whose spatial heterogeneity is difficult to describe deterministically. This motivated the use of a Monte Carlo approach to examine the effects of floodplain soils and their distribution on planform development, with the goal of including bank erosion properties in the analysis. Simulated bank erosion rates are controlled by the resistance to hydraulic erosion of the bank soils using an excess shear stress approach. The spatial distribution of critical shear stress across the floodplain is delineated on a rectangular, equidistant grid with varying degrees of variability. The corresponding erodibility coefficient is computed using a field-derived empirical relation. For a randomly disturbed distribution, in which the mean resistance to erosion exponentially increases away from the valley centerline, two relevant parameters are identified: the standard deviation of the critical shear stress distribution, which controls skewness and variability of the channel centerline, and the cross-valley increase in soil resistance, which constrains lateral migration and also affects bend skewness. For a purely random distribution, migrated centerlines exhibit larger variability for increasing spatial scales of floodplain soil heterogeneity. For equal stochastic variability of the corresponding governing parameters, relating meander migration to hydraulic erosion of the bank soils produces more variability and shape complexity than the “classic” bank migration approach of Ikeda et al. (1981), which relates migration rate to excess velocity at the outer bank. Finally, the proposed stochastic approach provides a foundation for estimating a suitable spatial density of measurements to characterize the physical properties of floodplain soils and vegetation.
 Ever since the seminal work of Ikeda et al. , numerous two-dimensional (2-D) depth-averaged analytical models have been derived to study the hydrodynamics, bed morphodynamics, and migration of meandering rivers [Blondeaux and Seminara, 1985; Crosato, 1987, 1989; Johannesson and Parker, 1989; Sun et al., 2001a; Zolezzi and Seminara, 2001] and the degree of planform shape complexity that can be obtained [Seminara et al., 2001; Frascati and Lanzoni, 2009]. Planform complexity may also result from the heterogeneity of the floodplain soils. Figure 1 shows some examples where floodplain heterogeneity affects river planform shapes through erodibility and soil mechanics properties (Beaver River in Alberta, Canada and Wabash River in Illinois and Indiana), vegetation dynamics (Mackinaw River in Illinois and Luangwa River in Zambia), and land use (Beaver Creek in Ohio). Examples in Figure 1show that, while most of the “signal” associated with floodplain heterogeneity is erased, over geological timescales, through cutoffs that govern the statistical long-term behavior of meandering rivers [Stolum, 1996; Camporeale et al., 2005], floodplain heterogeneity can affect short- and medium-term meander migration (i.e., migration over periods before cutoff occurrence), whose assessment is important for economic and social reasons (e.g., interaction with urban areas, prevention of damage to infrastructure, reduction of agricultural land losses, maintenance of biological diversity).
 While several studies have investigated the streamwise variation of bank erodibility and resulting bank erosion [e.g., Lawler et al., 1999; Hudson and Kesel, 2004; Wallick et al., 2006; Wynn and Mostaghimi, 2006; Constantine et al., 2009], the impact on planform shape has only lately been addressed systematically: in particular, the very recent contribution by Güneralp and Rhoads  has highlighted the importance of accounting for soil heterogeneity in river meandering models. Earlier work by Howard [1992, 1996] and Sun et al.  concentrated on the development of different features in the floodplain caused by river migration and their influence on the future development of the channel. In particular, Sun et al.  focused on the change in erodibility associated with features such as point bar, floodplain, and oxbow lake deposits. These researchers were able to reproduce the formation of meander belts over long time periods. Regarding planform shapes, Sun et al.  stated that the typical meander wavelength is mainly determined by flow rate, channel dimensions, and valley slope, and is independent of the different erodibility properties of the various sedimentary deposits. Perucca et al.  examined the role of vegetation on meandering river morphodynamics and the feedback between riparian vegetation dynamics and planform dynamics. Relating bank erodibility to the biomass density, they showed the possible occurrence of unusual meander shapes that do not show marked upstream skewness. Posner and Duan  studied planform evolution using (1) a migration model with a constant migration coefficient and (2) a model where the instantaneous migration coefficient was treated as a stochastic variable satisfying either a uniform or normal distribution. They found that the stochastic approach yielded more realistic predictions of meander planform evolution. Recently, Güneralp and Rhoads  examined how the scale, magnitude, and stochasticity of floodplain erosional resistance influence the planform evolution of meandering rivers. Using the power spectra of curvature series of migrated meander sequences, they showed that heterogeneity in erosional resistance has a major influence on meander evolution.
 The rate of lateral hydraulic erosion E* (m/s) for any bank or floodplain soil is modeled using an excess shear stress relation typically used for fine-grained materials [Partheniades, 1965]:
where M* is the erosion rate coefficient (m/s), τ*c is the critical shear stress (Pa), k* is the erodibility or detachment coefficient (m3/(N s)), and τ* is the shear stress acting on the soil particles (Pa). Variables with an asterisk are dimensional, whereas variables without an asterisk are dimensionless.
 The parameters M*, τ*c, and k* depend on soil properties and are therefore site and soil specific. The critical erosional strength denotes the cohesive strength provided by interparticle forces of attraction or repulsion acting at the microscopic level, including electrostatic forces, van der Waals forces, hydration forces, and biological forces [Papanicolaou et al., 2007]. Although the form of equation (1) is rather simple, complexity and uncertainty are introduced when assigning appropriate values for the erodibility parameters and the boundary shear stress τ* exerted by a given flow. These parameters are highly variable, which explains why observed rates of fluvial erosion range over several orders of magnitude [Darby et al., 2007]. Locally, values of the resistance to erosion parameters are time dependent, because of the effects of subaerial exposure, weathering, vegetation, or variations in soil water [Wynn et al., 2008]. Difficulties reside with the data collection itself: the commonly used jet erosion test [Hanson and Cook, 2004] cannot be sufficiently repeated, primarily because of time limits on site access resulting in small data sets to estimate the parameters that are needed [Constantine et al., 2009; Darby et al., 2010].
 In equation (1), τ* is the shear stress acting on the soil particles. Determining the bank shear stress is challenging. Several methodologies have been developed to calculate the shear stress in natural cross sections and implemented into one-dimensional (1-D) models [e.g.,Lundgren and Jonsson, 1964; Khodashenas and Paquier, 1999]. The effect of secondary currents on bank shear stress was studied for straight channels with trapezoidal cross sections [Knight et al., 2007; Rodriguez and Garcia, 2008] and cohesive river banks [Papanicolaou et al., 2007]. These findings still need validation for meandering rivers. Depending on planform shape and bed morphology, the distributions of bed and bank shear stresses near the bank toe could differ markedly from those in straight channels. Detailed measurements of shear stresses are available in the field. However, values mostly reflect specific cases and cannot be generalized. Lane proposed a methodology that relates bank shear stress to the bed shear stress through a proportionality factor. In bends, this parameter is influenced by secondary flow, bank slope, width-to-depth ratio, and difference in boundary roughness between bed and bank. Drag on small-scale topographic features can substantially alter the near-bank flow field and therefore the shear acting on the grains [Kean and Smith, 2006a], and depends on the magnitude of the shape parameters of bank topography [Kean and Smith, 2006b]. Bank form roughness may be a major component of the spatially averaged total shear stress [Darby et al., 2010]. Large woody debris also may represent a factor of further partition of the shear stress [Manga and Kirchner, 2000]. Note that modification of macroscale flow around slump blocks, and stress partitioning near the banks due to bank forms or woody debris can be taken into account indirectly by increasing τ*c in our approach and may also be associated with significant spatial variability in τ*c.
 Notwithstanding all these problems to quantify the erosion resistance parameters and the boundary shear stress, equation (1) has performed well for assessing stream bank erosion in incised streams in the midcontinental United States [e.g., Langendoen et al., 2000, 2009]. Resistance to erosion values were measured using a jet erodibility tester [Hanson and Cook, 2004] and boundary shear stress was computed using a one-dimensional flow model.
 The present study evaluates the impact of horizontal heterogeneity of floodplain soils on meander migration rates and patterns using a stochastic approach. The variability in meander planform development is determined as a function of the spatial variability in floodplain soil resistance to erosion and the spatial scale of this variability. The study extends the recent work by Güneralp and Rhoads , while introducing a relation for bank erosion computations which explicitly accounts for different erodibility properties associated with soil heterogeneity. For homogeneous soil, Motta et al.  showed that equation (1) produces greater planform shape complexity when compared to the classic approach in which the migration rate is based on excess velocity at the outer bank [Hasegawa, 1977; Ikeda et al., 1981]. The comparison between the two methods is herein extended to the case of heterogeneous floodplain soils. Two different spatial distributions for floodplain heterogeneity are used. Simulations are performed using a Monte Carlo probabilistic method to evaluate how the variability in erodibility parameters affects meandering river migration at the reach scale, analogously to the approach followed, at the local scale, by Parker et al.  and Samadi et al. for the computation of the factor of safety for planar and cantilever failure, respectively. This work identifies to what extent meander migration is affected by floodplain heterogeneity, the parameters that govern this effect, and how their variability affects migrated-centerline variability. Furthermore, study outcomes could be useful to determine feasible space and time scales for deterministic simulations of bend evolution when, for instance, there is interest in remeandering a stream for restoration and naturalization purposes.
where s* and n* are streamwise and transverse coordinates, U* and V* are streamwise and transverse depth-averaged velocities,g* is the gravitational acceleration, H* is water stage, τ*s and τ*n are streamwise and transverse bed shear stresses, ρ* is water density, D* is water depth, and C* is the local channel curvature. The bed morphology is described by assuming that the transverse bed slope is proportional to the local curvature through a constant of proportionality A, named the scour factor by Ikeda et al. :
where η1∗ is the bed elevation deviation from the bed elevation of the channel centerline at a particular cross section and Dch∗is the reach-averaged water depth for uniform flow. As reported byMotta et al. , an analytical solution of the linearized equations (2)–(4), for low curvature, constant discharge Q and channel width 2B*, is obtained for streamwise and transverse velocities and water depth. The solution is expressed as the sum of their uniform flow reach-averaged value (indicated by subscriptch) and a perturbation (indicated by subscript 1), which varies in both the streamwise and transverse directions. The solution depends on upstream and local curvature, sinuosity, half width to depth ratio β = B∗/Dch∗, reach-averaged Froude numberFch, reach-averaged friction coefficientCf,ch, and the scour factor A.The solution for the streamwise depth-averaged flow velocity is
where the variables have been normalized as U = U∗/Uch∗, s = s∗/B∗, n = n∗/B∗, and C = B∗C∗. The parameters , , , and are functions of β, Fch2, Cf,ch, A, and velocity and curvature at the upstream boundary (see Garcia et al.  and Motta et al.  for details). Following Engelund and Hansen , the friction coefficient Cf,ch is calculated as
where d*s is a measure of the bed grain size. Here, the friction coefficient is assumed to be constant everywhere in the channel for a given channel planform configuration (i.e., a given Dch∗).
 The streamwise bed shear stress is then calculated as
 Two alternative methods are implemented in the RVR Meander platform for the computation of bank erosion and centerline migration. The first method is a migration coefficient method (named MC method hereafter). The outer bank displacement rate equals the centerline displacement rate (maintaining constant channel width), and is proportional to the excess velocity at the outer bank through a dimensionless migration coefficient [Hasegawa, 1977; Ikeda et al., 1981]:
where R* is the rate of migration (with dimensions of length over time) and U*bis the depth-averaged near-bank velocity. The dimensionless migration coefficientE0 is usually obtained via calibration against historic channel centerlines.
 In the second, physically based, method for bank erosion (named PB method hereafter), the centerline displacement rate is equal to the displacement rate of the outer bank. The outer bank is defined as the bank that experiences more erosion [Motta et al., 2012] as estimated by equation (1). Therefore, the terminology “physically based” refers to the way bank erosion is modeled. In terms of channel width evolution, we still assume constant channel width in space and time. While generally being supported by empirical observations [Ikeda et al., 1981], this assumption is a mathematical and physical simplification to obtain the analytical solution given by equation (6) and is not a result of modeling conservation of sediment mass. All applications shown in this paper adopt the PB method, except for the case when MC and PB methods are compared for the case of heterogeneous floodplains in section 4.5.
3. Representation of Floodplain Soils
3.1. Generation of Resistance to Erosion Values
 The spatial distribution of erosion resistance across the floodplain can be highly variable, and depends on sediment redistribution, soil development from the evolution of the fluvial system through channel migration and overbank deposition, vegetation dynamics, land use, and anthropogenic modifications. Quantitative information is usually limited. For instance, in the United States, the USDA National Resources Conservation Service (NRCS) soils database (http://soildatamart.nrcs.usda.gov/) includes information on soil erodibility, but is described in terms of RUSLE parameters [Renard et al., 1997]. Database fields are available for critical shear stress and other parameters, however these fields are usually empty due to the lack of resources for field observations.
 In our study the resistance to erosion properties of the floodplain soils are represented on a rectangular, equidistant grid characterized by grid spacing Δx* and Δy* in the x* and y* direction, respectively. The parameters M* (or k*) and τ*c (for PB method) or E0 (for MC method) are stochastically assigned at each grid node. Two distributions are used to generate the values of critical shear stress τ*c (or E0, in case of the MC method) at floodplain grid nodes. The first distribution is purely random, which represents the situation of laterally unconfined valleys containing soil anomalies. Such anomalies can be caused by: spatial differences in clay-silt content [Schumm, 1960; Hooke, 1980]; variations in particle size, bulk density and ionic strength [Bull, 1997]; or weathering, physical, biological, and chemical processes [Wood, 2001]. Furthermore, irregularities in the modern floodplain can be remnants of the natural meandering evolution and floodplain reworking over geologic time scales, Parker et al. distinguish between microscale variability (at the order of one cross-section bank) and mesoscale variability (at the order of several cross sections). We focus here on the latter, larger scales, which is discussed below. The second distribution is characterized by decreasing erodibility in the cross-valley direction, which represents (1) a diffusion-dominated floodplain environment where sediment size and consequently erodibility decrease away from the channel [Pizzuto, 1987] or (2) a situation where cutoff deposits tend to confine the lateral extent of migration across the valley. This confined region is known as the meander belt [Allen, 1965; Sun et al., 1996].
 The latest version of RVR Meander (available at http://rvrmeander.org/) can treat the case of user-defined heterogeneous floodplain as described above.
3.1.1. Purely Random Distribution
 The first method employs a purely random (PR) distribution, analogously to Güneralp and Rhoads . Values of τ*c (or E0) are generated from a normal distribution (μ, σ), where μ and σare mean and standard deviation respectively, using the Box-Muller transform [Box and Muller, 1958] in the polar form [Bell, 1968; Knopp, 1969]. The Gaussian distribution was selected in analogy to the work by Güneralp and Rhoads , who assumed, as first approximation, a Gaussian distribution of the migration coefficient E0. Furthermore, studies on the natural variability of geotechnical parameters at bend scale [Parker et al., 2008] and reach scale [Samadi et al., 2011] have shown that Gaussian fits of parameter distributions are statistically significant. In particular, Parker et al. demonstrated, using the D'Agostino-Pearson statistical test for normality, that sampling variability may be in general the only factor responsible for any deviations from a Gaussian distribution for cohesion, friction angle, and saturated unit weight;Samadi et al. observed that values of coefficient of determination of data-fitted Gaussian probability density function curves for cohesion and matric suction angle are close to unity. Therefore we can reasonably expect to have a similar Gaussian distribution for critical shear stress since, like cohesion,τ*c also represents the bond strength between particles. Unfortunately, even the largest database of τ*c and k* known to the authors [Simon et al., 2010] does not have sufficient spatial resolution to determine the statistical distribution of these parameters in the field. Because of physical consistency, whenever a negative random value is generated, it is regenerated until it is positive. While regeneration introduces a skewness in the normal distribution, its magnitude is negligible, since the probability associated with negative values is very small (about 10−3 or smaller) for the values of μ and σ adopted here. Therefore regeneration happens very rarely.
 The length scale over which soil property variations occur can be expressed in terms of channel widths [Sun et al., 1996] or channel centerline wavelength or amplitude [Güneralp and Rhoads, 2011], which themselves can be related to one another using commonly used, empirically derived equations [e.g., Williams, 1986]. As mentioned above, herein we ignore microscale variability. The focus is on mesoscale variability, which is modeled by length scales equal or greater than one channel width; features produced by migration of entire bends correspond to length scales in the order of one meander wavelength as assumed in section 4.3.
3.1.2. Randomly Disturbed Distribution
 The second method is a randomly disturbed (RD) distribution, where the mean value of the critical shear stress increases exponentially as a function of the distance l from the valley centerline
where bis the rate of increase in the cross-valley direction. The exponential form is analogous to that derived analytically byPizzuto to model cross-valley grain size trends associated with overbank flows and to that adopted bySun et al.  for the temporal increase of the erodibility in oxbow lakes because of silt infilling (note that time and distance from valley centerline are correlated). At each floodplain grid node, the actual value of critical shear stress is then generated using a Gaussian distribution characterized by a mean given by equation (11) and an assigned standard deviation σ.
3.1.3. Assignment of Erodibility Parameters
 After τ*c is assigned at a floodplain grid node, the corresponding value of erodibility k* in m3/(N s) is computed using the expression developed by Hanson and Simon  for channels in the loess areas of the midcontinental USA and also recently adopted by Darby et al. :
which can be rewritten in terms of erosion rate coefficient in m/s as
where c = 0.2 × 10−6 m2/(N0.5 s).
 Note that in general, measured values of k* and M* do not necessarily satisfy equation (12) or (13). Scatter of k* over about 1 order of magnitude is present in the data collected by Hanson and Simon from which the equations were derived. Applications at the cross-section scale have highlighted the need to calibrate the erodibility coefficient to obtain the best agreement between calculated and measured bank retreat rates [Darby et al., 2007; Rinaldi et al., 2008]. Calibrated values are within the range of the scatter in k*. In the case of meander migration modeling, proper calibration also depends on the numerical smoothing of curvatures and channel centerlines [Crosato, 2007]. Equation (13) provides a functional relation that correlates M* and τ*c, for the purpose of generating erosion resistance distributions used in the Monte Carlo simulations. We later discuss the impact of deviations in M* from equation (13) on meander migration rates and shapes.
3.2. Estimating Erosion Resistance Parameters at an Arbitrary Location
 The value V* of the resistance to erosion properties at an arbitrary location (x*, y*) is computed by inverse distance weighting (IDW) interpolation [Shepard, 1968]:
where (xi∗, yi∗) are the coordinates of each vertex of the grid cell containing the arbitrary point (x*, y*), and p is a positive real number. In our study, p equals 1. A sensitivity analysis showed no major impact of p on meander migration.
 In summary, for the PB migration model, values of τ*c and M* are assigned at floodplain grid nodes, where they satisfy Hanson and Simon's  relation (13). Inside the floodplain grid cells, both values of τ*c and M* are obtained through IDW interpolation of the corresponding values at the vertices of their cell, using equation (14). As a consequence interpolated values of τ*c and M* do not exactly satisfy equation (13), though the deviation is small. It was verified that the selected procedure does not have any significant impact on simulated meander migration. Also note that if we instead interpolated τ*c and k* and computed M* as M∗ = k∗τc∗, we would get a slightly different value than that obtained from direct interpolation of M*. Again, it was verified that the effects of this procedure on meander migration are very small.
 Our approach of assigning resistance-to-erosion values of floodplain soils at grid nodes is different from that adopted byGüneralp and Rhoads , who assigned resistance-to-erosion values at cells (patches). A grid of point values instead of patches is similar to a data set of erosion resistance parameters obtained from a field campaign to characterize floodplain soils. While floodplain soil properties at grid nodes are random, values at an arbitrary location (x*, y*) are obtained through interpolation using equation (14) and are therefore deterministic. Hence, the grid cell size is an indicator of the spatial variability of the floodplain soil properties.
 As the river migrates across the floodplain, the parameters τ*c and M* (PB approach) and E0 (MC approach) along the right and left bank are computed at the beginning of each migration time step using equation (14). The size of the floodplain grid is set large enough to contain the possible migration of the channel centerline. In the case of the PB approach, the potential erosion distance is computed at both banks (according to the corresponding shear stress and critical shear stress at that bank) and the greatest value determines the direction of migration and is taken as the meander migration distance. In the case of the MC approach, E0 at left bank is used if the positive excess velocity is at the left bank, otherwise E0 at the right bank is adopted.
4. Test Cases
 The impact of the horizontal heterogeneity of floodplain soils on rates and patterns of meander migration is evaluated for various distributions of soil erodibility and two initial channel alignments: a sine-generated channel and the Mackinaw River, Illinois.Table 1reports the run parameters used to simulate hydrodynamics and bed topography for the two channel alignments. For convenience, the value of Manning's roughness coefficient corresponding to the value of the friction coefficient is also listed. Because reach-averaged Froude number, friction coefficient, and half-width to depth ratio change over time as the reach-averaged depth varies with changing sinuosity,Table 1 reports the values corresponding to a straight channel of the same width and having the valley slope S0.
Table 1. Run Parameters for the Two Channel Alignments Useda
Here Q is discharge, 2B* is channel width, S0 is valley slope, F0 is Froude number for a straight channel with slope S0, Cf0 is friction coefficient for a straight channel with valley slope S0, β0is half-width to depth ratio for a straight channel with slopeS0, A is transverse bed slope parameter (scour factor), and n0 is the Manning's roughness coefficient for a straight channel with slope S0.
Mackinaw River, Illinois
 To assess the variability in erosion resistance properties of floodplain soils, which are generated stochastically, the Monte Carlo method was used. Different distributions of the erodibility properties of the floodplain soils were generated repeatedly, and the corresponding channel migration was simulated. Each simulation period was smaller than the time to cutoff, such that all planform complexity is caused by floodplain heterogeneity and not by cutoff dynamics. Note that reported simulation periods and time steps are scaled, and can be converted to actual times using the intermittency of the modeling discharge.
4.2. Simulated Meander Migration for RD Floodplain Soil Distribution
 Monte Carlo simulations for RD floodplain soil distribution were carried out for a sine-generated channel aligned parallel to the valley centerline. The channel centerline is expressed as
where θ is the angle between centerline and valley axis, θ0 is its value at the crossover point, and Λ is the length of the channel centerline over one meander wavelength. A 2040 m long (2000 m long sinuous section with 20 m long straight entrance and exit sections) and 30 m wide channel with Λ* = 250 m and θ0 = 55° is considered, which corresponds to a relatively low sinuosity of 1.27. Two hundred equally spaced nodes describe the initial centerline, yielding a node spacing of about 10.2 m. The channel contains 17 bends. The centerline evolution was simulated for a 20 year period employing a time step of 0.2 year. Table 1 lists the various simulation parameters (Channel 1).
Table 2 reports the floodplain soil characteristics of six Monte Carlo simulations using the RD distribution, which are represented by two different values of b(indicator of the cross-valley increase of soil resistance) and three different values ofσ (indicator of the local variability in soil properties). In particular, values of σwere selected to cover the range between quasi-uniform erodibility distribution in the down-valley direction (lowσ, see examples in Figures 2c and 2d) and highly variable erodibility distribution in the down-valley direction (highσ, see examples in Figures 2e and 2f). Table 2 indicates the critical shear stress variability at the valley centerline, where variability is hereafter defined as two times the coefficient of variation, that is [Samadi et al., 2011]. The grid cell size Δx* = Δy* = 60 m was selected to produce noticeable variability of the migrated centerlines (see section 4.3for in-depth analysis of the effect of the grid cell size). This cell size is equal to two times the channel width, almost two times the amplitude of the initial centerline, and about one third of the wavelength of the initial centerline measured along the valley axis. One of the rows of the floodplain grid nodes coincides with the axis of the initial centerline aty* = 34.4 m. Five hundred floodplain soil distributions were generated for each RD distribution. It was verified that 500 simulations were enough to compute robust and repeatable statistics and to describe the envelope of migrated centerlines.
Table 2. RD Floodplain Soil Distribution Parameters for Monte Carlo Simulations of a Sine-Generated Channel
Grid Size (m)
Figure 2 shows an example realization of a generated floodplain soil spatial distribution and all the simulated final channel centerlines for the six test cases. Generally, higher values of σ lead to higher variability in centerline migration. Furthermore, higher values of b produce less lateral migration and a more pronounced tendency toward upstream skewness.
 Following the approach of Posner and Duan , statistics were computed using the (x*, y*) coordinates of the centerline nodes at the end of each simulation on a bend-by-bend basis, taking advantage of the configuration selected for initial channel centerline and valley axis. To quantify lateral migration, the mean value of the coordinatey* of a meander lobe is used
where n is the number of centerline nodes describing a lobe. The streamwise orientation of a lobe is described by the skewness of the coordinate x*
where is the standard deviation of the coordinate x*
A positive (negative) value of represents upstream (downstream) skewness. Whereas Posner and Duan  used these statistics to compare simulated and measured bend migration, here the statistics are used to describe and compare the migrated centerline variability associated with different floodplain soil distributions.
Figure 3 compares the median values of (Figure 3a), the interquartile range (difference between 75th and 25th percentiles) of (Figure 3b), the median value of (Figure 3c), and the interquartile range of (Figure 3d) for the lobes of the final centerline. Note that the absolute values of are actually plotted in Figures 3a and 3b.
 For increasing b(indicator of cross-valley increase in soil resistance), the value (indicator of lateral migration) generally decreases (Figure 3a). Furthermore, high bvalues enhance the tendency toward upstream-skewed bends which is clearly shown in Figure 3c. This is the result of constrained lateral migration. A limit condition of lateral constraining is analyzed in Figure 4, where nonerodible valley walls are modeled by equating E0 and M* locally to zero in the MC and PB approach, respectively. However, the floodplain soils are still erodible; their resistance to erosion is increasing linearly in longitudinal direction. Upstream-skewed bends develop as a result of the lateral constraining, while longitudinal heterogeneity affects longitudinal migration in the erodible corridor. Note that the PB approach reproduces more closely the shapes typically observed in confined meandering rivers such the Beaver River in Canada (Figure 1a).
 The parameter σ (indicator of the local variability in soil erodibility) does not significantly affect as long as its value is not too high. In this case, patches of less erodible soil constrain lateral migration. Upstream skewness is reduced for larger σ (Figure 3c). However, the variation in both and increases for larger values of σ (Figures 3b and 3d), which means that a broader spectrum of planform shapes can be obtained both in terms of lateral migration and bend skewness.
4.3. Effect of the Length Scale of Floodplain Soil Heterogeneity
 To better appreciate the spectrum of migrated planform shapes and the effect of length scale of floodplain soil heterogeneity, Monte Carlo simulations were made for a PR floodplain soil distribution and a natural channel alignment. The alignment considered is a 4 km long reach of the Mackinaw River in Illinois. The reach is located in Tazewell County about 15 kilometers upstream of the junction of the Mackinaw River with the Illinois River between the towns of South Pekin and Green Valley. The 1951 historic centerline was taken as starting centerline for our tests.
 The initial channel centerline was represented by an equidistant set of 350 nodes with a spacing of 11.4 m. Channel width is 38 m and valley slope is 0.0009 m/m [Motta et al., 2012]. The mean sinuosity of the study reach is 1.34. The model flow discharge was selected as 46.2 m3/s, which corresponds to the 10.4% duration discharge according to the daily flow record at USGS station 05568000 near Green Valley. It was determined by trial and error to be the discharge that captures the increasing sinuosity pattern observed in the Mackinaw River after 1951. The upstream model boundary was set in a straight reach. Because the transverse velocity distribution at the upstream boundary is not known, it was assumed constant. The centerline evolution was simulated for a 40 year period and a time step of 0.2 year. Further, a second-order Savitzky-Golay filter (averaging window of 5 nodes and applied every 10 iterations) was used to smooth the simulated channel centerline. SeeMotta et al.  for details of this method. Table 1 reports the various simulation parameters (Channel 2).
 Monte Carlo simulations were performed for different values of grid cell size (Δx*, Δy*) (i.e., different length scales of floodplain heterogeneity). The grid size ranged from one to ten times the channel width. The greater grid cell size was selected to obtain patches of substantially uniform material with spatial dimensions in the order of the dominant arc wavelength of the initial centerline (about 450 m). A value = 9 Pa was adopted, as used by Motta et al.  for simulating the migration of the Mackinaw River reach in homogeneous floodplain soils, to account for slump block bank protection and vegetation. Two values were selected for σ, 1 Pa (Figure 5) and 2.25 Pa (Figure 6), which correspond to to 22% and 50%, respectively. The latter case allows for modeling dramatic spatial gradients of floodplain soil properties that can occur in natural floodplains at clay plugs, i.e., former oxbow lakes filled with fine-grained alluvial deposits [Fisk, 1947]. One thousand floodplain soil distributions were generated for each Monte Carlo simulation, which allowed for computing robust migrated-centerline envelopes and statistics.Table 3 summarizes the combinations of parameters adopted.
Table 3. Mackinaw River, Illinois, Test Cases to Examine the Effects of the Length Scale of Floodplain Soil Heterogeneity on Migrated Centerline Variability VCL
Δx* = Δy* (m)
Δx*/(2B*) = Δy*/(2B*)
VCL for = 0.22
VCL for = 0.50
Figures 5 and 6 show that smaller grid cell sizes are associated with lower variability in centerline migration. Variability of the migrated centerlines VCL is defined here as the ratio between the area occupied by all migrated centerlines (whose bounds were manually digitized) and the length of the migrated centerline obtained for a homogeneous floodplain characterized by mean floodplain property values (τ*c = 9 Pa, M* = 6.0 × 10−7 m/s), and normalized by the channel width 2B*. VCL increases less than linearly for increasing grid cell size (Table 3 and Figure 7). Figure 7 shows that the relation between length scale of heterogeneity and centerline variability can be represented, at least for these case and erodibility parameters, by an exponential decay formula, which suggests that there is an upper limit for VCL. The decreasing rate of increase of channel centerline variability with floodplain heterogeneity length scale is due to two reasons. First, the existence of larger patches of floodplain characterized by low resistance to erosion, which are present for larger grid cell sizes. In other words, while large patches of easily erodible material allow more meander migration and lead to the increasing pattern in Figure 7, the coexistence of large patches of hardly erodible material limits this trend, which becomes less than linear. The second reason is the reduction of shear stresses resulting from an increase in channel sinuosity and a decrease in channel slope. A threshold approach for bank erosion as in equation (1) is associated to decreasing bank erosion rates in time for increasing sinuosity. The impact of this reduction depends on the magnitude of the critical shear stress. For the quite high values of τ*c used here to indirectly account for the effect of slump block bank protection and vegetation, this effect is marked. For longer time scales, this pattern of shear stress reduction is counteracted by periodic bend cutoff, which rejuvenates channel activity [Hooke and Yorke, 2010] and again increases overall flow velocities and shear stresses. Up to that point, however, the preferential occupation of highly erodible sections of floodplain by the channel (as in the examples in Figure 8) and simultaneous reduction of slope and shear stresses yield lower erosion rates. Increasing the critical shear stress variability enhances centerline variability (Figure 6) but does not affect the pattern of the relation between heterogeneity length scale and centerline variability (Figure 7). Therefore, our analysis generalizes and extends the conclusions reached by Güneralp and Rhoads  regarding the increasing impact of floodplain heterogeneity for increasing heterogeneity length scale and magnitude of stochastic variability of erodibility. Furthermore, by using the formulation given by equation (1) for bank erosion, we found that this trend is not indefinite but mitigated by the river evolution itself.
4.4. Complexity of Planform Shapes
 While the value of (or, equivalently, σ) does not affect the pattern of the relation between floodplain heterogeneity length scale and VCL, it does affect planform shapes. The model simulates shapes different from the upstream-skewed features, whichGarcia et al.'s  model for hydrodynamics and bed morphodynamics produces in subcritical conditions in homogeneous soil [Seminara et al., 2001], especially for large floodplain heterogeneity length scales. Figure 8 shows examples of migrated centerlines obtained for case PR10 with = 0.50 (see Table 3). More complex shapes can be observed, such as downstream-skewed bends, compound loops, elongated bends, and compound bends.
 Though the diversity of planform shapes that can be obtained in heterogeneous floodplains has been previously studied using a one-parameter MC approach [e.g.,Güneralp and Rhoads, 2011], the two-parameter PB method allows for identifying another factor contributing to planform complexity, namely, the relation betweenτ*c and M*. In the cases shown so far erodibility at floodplain nodes was based on relation (13). However, that relation was empirically derived through regression analysis by Hanson and Simon . Actual values of M* can differ as much as several orders of magnitude due to variations in soil water content and its chemistry, or organic content [e.g., Hanson and Hunt, 2007; Wynn et al., 2008]. Figure 9 exemplifies the effect of such deviation, where the spatial distribution of erosion rate coefficient M* was generated at floodplain grid nodes either using equation (13) (referred to as M*HS) and 1.5 and 0.5 times M*HS. For larger values of M*, migration distance increases and there is a greater tendency toward downstream skewed bends (see lobes L1, L2, and L3 in Figure 9), which is caused by higher erosion rates in the areas of the floodplain where the shear stress at the bank exceeds the critical shear stress.
4.5. Comparison of MC and PB Approaches for Heterogeneous Floodplain Soils
 The effect of MC and PB migration approaches on migrated-centerline variability was examined using PR distributions of the parametersE0 (for the MC approach) and τ*c or M* (for the PB approach). Test cases are summarized in Table 4. Mean values of E0, τ*c, and M* were selected to produce comparable reach-averaged migration distances. In the case of the PB approach, for randomly generatedτ*c values, the corresponding values of M* were computed using equation (13), whereas for randomly generated M* values, the corresponding values of τ*c were computed using the inverse of equation (13). A grid size Δx* = Δy* = 2B* = 38 m was used and a period of 20 years was simulated. For this analysis it was verified that 500 simulations for each Monte Carlo simulation were enough to compute robust statistics and describe the envelope of migrated centerlines.
Table 4. Resistance to Erosion Values Used in the Test Cases to Compare the Variability in Channel Centerline Migration Produced by the MC and PB Methods
5.0 × 10−7
5.6 × 10−8
5.0 × 10−7
1.3 × 10−7
5.0 × 10−7
2.3 × 10−7
6.0 × 10−7
6.7 × 10−8
6.0 × 10−7
1.5 × 10−7
 Two kinds of comparisons between MC and PB approaches were made: (1) comparison of migrated centerline variability VCL for two values of variability (V equals 22% and 50%) in E0, τ*c, or M*; and (2) comparison of centerline complexity for similar VCL.
 With respect to comparison 1, Figure 10 and Table 4 show that the PB method produces greater variability in channel centerline migration than the MC method. The variability VCL in the PB approach is between 3.2 to 5.9 times greater than that of the MC approach for the cases examined herein.
 To compare the MC and PB approaches for similar VCL, an additional Monte Carlo simulation was performed (case E03 in Table 4), which produces a VCL similar to case PBtau1. We computed, following the approach of Güneralp and Rhoads , the power spectra of the curvature series of all migrated centerlines for both Monte Carlo simulations E03 and PBtau1 (Figure 11). Both the MC and PB approaches produce spectra with one or more major low-frequency peaks and many minor high-frequency peaks, which, as observed byGüneralp and Rhoads , is typical of natural rivers. However, the PB approach is able to produce features with higher dimensionless frequency (minor peaks with dimensionless frequency greater than 0.1 in Figure 11) than the MC approach for similar VCL. In other words, the PB approach is associated with greater planform complexity, confirming for heterogeneous floodplain what was already observed by Motta et al.  for homogeneous floodplains. Comparisons 1 and 2 also show that meander migration patterns obtained with the PB method cannot be replicated by using the MC method, no matter how the variability of the migration coefficient E0 is increased.
 Comparing the MC and PB migration approaches, the former is linear in the velocity at the bank (equation (10)), while the latter is quadratic (the fluvial erosion rate is proportional to the shear stress, equation (1), which depends on the squared velocity, equation (9)). However, Xu et al.  showed that this difference cannot explain the observed increase in migrated centerline variability and planform complexity, which are mainly caused by the presence of an erosion threshold in the PB approach instead. The MC approach without an erosion threshold (meander migration always occurs as long as there is an excess velocity at the outer bank, equation (10)) predicts a smooth migration pattern. Further, the increased migrated-centerline variability for the PB approach is also caused by the use of two parameters,τ*c and M* (even if correlated by equation (13)), whereas the MC approach only uses one, E0.
 The presented work can be used to determine characteristic space and time scales required to perform deterministic simulations of meander migration before cutoff occurrence. Our stochastic methodology may provide a foundation for determining a suitable spatial density of measurements needed to characterize the physical properties of floodplain soils and vegetation. The spatial density can be represented by the grid cell size, which is an indicator of the floodplain heterogeneity length scale. Being our PB approach for bank erosion based on physical parameters, soils samples and information on vegetation distribution can help specify an appropriate value of whereas it would be difficult to define a priori a suitable value of for the MC approach. For low variability in erosion resistance properties of the soils, the envelope of migrated centerlines is a relatively narrow corridor (for instance, see Figures 2a, 2b, 2c, 2d and 5a). For spatially highly variable floodplain soil distributions (for example, where multiple cutoffs have historically happened or for a nonuniform distribution of vegetation), it is important to identify the highly and hardly erodible floodplain sections. Hence, a finer resolution of floodplain soil properties reduces the variability in the predicted migration of the channel centerline. Because the increase in centerline variability with increasing length scale of heterogeneity is less than linear, curves similar to those in Figure 7 may assist in determining the spatial density of field measurements.
 Other factors may contribute to planform complexity, such as hydrodynamics and bed morphodynamics [Seminara et al., 2001; Frascati and Lanzoni, 2009]. In this study we assumed steady bed morphology (for a given planform configuration), which is related to the local curvature (equation (5)), instead of explicitly simulated by solving the sediment mass conservation equation [Garcia, 2008]. More sophisticated meander migration models are available, such as the linear models of Crosato [1987, 1989], Johannesson and Parker , Zolezzi and Seminara , and Luchi et al. , or the fully nonlinear models of Abad et al.  and Bolla Pittaluga et al. . Camporeale et al.  showed that models which include the free response of the bed sediments can give rise, or wash away, multilobed planimetries in homogeneous floodplain. Seminara et al. demonstrated that meandering streams display alternating upstream- and downstream-skewed loops as well as multiple loops in the super-resonant regime without accounting for cutoff processes, spatial heterogeneity of bank erodibility or human constraints. Migrating bed forms [Abad and Garcia, 2009] as well as changes or seasonal variability in watershed hydrology can also impact meander migration. The importance of all these factors on producing complex meander channel planform, relative to the distribution of erosion resistance properties of floodplain soils, could be studied further by incorporating the above mentioned linear and nonlinear models into the physically based migration modeling system presented herein.
 A stochastic analysis of the impact of the horizontal heterogeneity of floodplain soils on rates and patterns of migration of meandering streams was performed. Meander migration was caused by fluvial erosion of the stream banks. The hydrodynamics and bed topography of the meandering stream were simulated using a simple Ikeda et al.  type model. Three parameters were analyzed: the local variability (or randomness) in soil resistance to erosion (σ), the cross-valley increase of soil resistance (b), and the spatial scale of heterogeneity represented by the cell size of the grid used to delineate the floodplain soils.
 The analysis most importantly showed that floodplain soil complexity contributes to planform complexity (e.g., elongated bends, symmetric compound loops, and compound bends), and can produce downstream-skewed lobes, which anIkeda et al.  type model cannot produce in a homogeneous floodplain. The analysis also highlighted three key parameters and the nature of their impact on river migration in heterogeneous floodplains. The parameter σmainly controls bend shape. Larger values increase the possibility of occurrence of downstream-skewed bends and generally produce a broader range of bend skewness values. The parameterb mostly controls the extent of lateral migration. However, higher b values are associated with more pronounced upstream skewness of bends. A smaller grid cell size, i.e., finer scale of soil heterogeneity, results in lower variability in simulated channel centerline migration, confirming what was found recently by Güneralp and Rhoads using a MC approach for migration. Further, the increase in variability with increasing grid cell size is less than linear. Finally, comparison of stochastic MC and PB migration methods showed that the PB approach is intrinsically associated with greater migrated centerline variability and planform complexity, primarily because of the use of a two-parameter erosion relation in combination with an erosion threshold to characterize bank migration.
 This research was supported by an agreement from the U.S. Department of Agriculture, Forest Service, Pacific Southwest Research Station and using funds provided by the Bureau of Land Management through the sale of public lands as authorized by the Southern Nevada Public Land Management Act. This work was performed under Specific Cooperative Agreement 58-6408-8-265 between the Department of Civil and Environmental Engineering at the University of Illinois at Urbana-Champaign and the U.S. Department of Agriculture, Agricultural Research Service, National Sedimentation Laboratory. Jorge D. Abad's participation was supported by his academic start-up funding provided by the Department of Civil and Environmental Engineering of the University of Pittsburgh. Eric Waschle Prokocki is also acknowledged for the interesting discussions on floodplain soil patterns. Several comments and constructive criticism by four reviewers helped to improve this contribution. Their help is herewith acknowledged.