We assume a simple working framework in order to limit model complexity and the number of parameters. The erosion rate

$\epsilon $ (m s

^{−1}) at the streambank toe is modeled using the excess shear stress equation [

9,

26,

27,

29,

58]:

where

${k}_{d}$ is an erodibility coefficient (m

^{3} N

^{−1} s

^{−1}),

${\tau}_{a}$ is the boundary shear stress applied by the flow (Pa) at the streambank assuming the cross section is trapezoidal,

${\tau}_{c,veg}$ is critical shear stress considering the mechanical effects of roots (Pa) and

$a$ is a dimensionless empirically derived exponent. It is usually assumed that

$a$ takes values close to 1 [

27,

59].

${k}_{d}$ can be estimated as:

where

$c$ is a coefficient usually ranging between 0.1 and 0.2 for cohesive material. Since BankforNET uses the excess shear stress equation to calculate erosion rates not only for cohesive but also for noncohesive material,

$c$ is adapted empirically based on

${\tau}_{a}$ and median particle diameter

${D}_{50}$ (mm) for noncohesive material. Mean applied hydraulic shear stress [

3,

46] considering mean bend radius

$r$ (m) assuming uniform flow is calculated as [

60]:

where

${\rho}_{f}$ is fluid density (kg m

^{−3}),

$g$ is the gravitational acceleration (9.81 m s

^{−2}),

${R}_{h}$ is the hydraulic radius (m),

$S$ is mean channel slope (m m

^{−1}) and

$\overline{b}$ is mean channel width (m).

${R}_{h}$ is derived using the Gauckler–Manning–Strickler equation, where Strickler’s roughness coefficient is derived empirically as a power function of

${D}_{50}$ (mm) for every cross section. To characterize the erodibility of streambank material, critical material dependent shear stress

${\tau}_{c}$ (Pa) is estimated using Shields criterion [

61,

62]. Rearranging the equation, critical shear stress can be formulated as:

where

$\theta $ is the dimensionless Shields entrainment parameter,

${\rho}_{s}$ is solid density (kg m

^{−3}) and

${D}_{50}$ is median grain size (m). Reported values of

θ scatter between 0.012 and 0.3 [

62] and are divided into classes based on the particle size classification.

θ hereby defines the threshold at which sediment particles are at incipient motion for noncohesive materials [

3], but this approach has also been applied for cohesive materials in few cases [

63]. As

θ depends on the sediment diameter, it can be classified based on measured

${D}_{50}$ values. Depending on

${D}_{50}$, BankforNET uses a fitted normal distribution function to determine random values of

θ (

Figure 1) within the range of permissible

θ values for each particle size class, where the upper and lower threshold are based on values found in the literature [

3,

38,

62,

64,

65]. For example, if we have coarse gravel (with

${D}_{50}$ = 27 mm, as presented in

Figure 1), the lower and upper permissible threshold of

θ range from 0.044 to 0.052 based on this defined particle size class [

64]. During the iteration process, 10,000 possible values for

θ are randomly generated considering a normal probability distribution. The mean value is defined as a function of

${D}_{50}$ and the standard deviation of the normal distribution is defined based on the particle size class and the corresponding upper and lower threshold for

θ. Subsequently, 10,000 possible

${k}_{d}$ values are computed resulting in a total of 10,000 ε values. The final modeled erosion rate then represents the mean cumulative erosion computed from all 10,000 iterations. Although we are not aware of any comprehensive data collection for the characterization of the distribution, the implementation of a normal probability distribution for a permissible and feasible range of

θ is important to emphasize how the modeled results are influenced by the estimation of

θ.

The mechanical effects due to the presence of roots in the soil are implemented as an additional term in the estimation of

${\tau}_{c}$. The modified critical shear stress including the effects of roots

${\tau}_{c,veg}$ is calculated using the root area ratio (RAR). The RAR is the ratio of total root cross sectional area divided by the total area of the soil profile in which the plant grows [

66]. Adapting the equation proposed by Pasquale and Perona [

42],

${\tau}_{c,veg}$ is calculated as:

where

$a$ =

$3\times 10{}^{-4}$,

$b$ =

$9\times {10}^{-3}$ and

V_{s} = 2.4 represents the soil volume of the plot that was used to calibrate the equation [

42]. The work from Pasquale and Perona [

42] studied the effects of roots on streambed erosion and found that local hydrodynamic bed shear stress conditions when exceeding some critical value gradually cause erosion. This can ultimately lead to uprooting and subsequent entrainment of vegetation. The change in local hydrodynamic bank shear stress conditions also causes streambank erosion when the critical value is exceeded. Even though the uprooting and entrainment process for roots situated on the bed or the streambank may be different, we assume that the effects of roots affecting critical shear stress are the same for both processes.

Measuring RAR in the field is a time-consuming task. Since BankforNET is intended to rapidly assess areas at risk of hydraulic bank erosion considering the effects of roots, RAR is estimated using an adapted root distribution model presented in Schwarz et al. [

67], additionally considering vertical root density as proposed by Tron et al. [

68]. The root distribution model uses tree stem diameter at breast height (DBH) to estimate root density and maximum rooting distance from the tree stem. The essential root diameters are calculated for each distance from a tree as an upper boundary for root diameter distribution. The number of roots and the values of fine and coarse roots are calculated based on empirical root distribution data. In this framework, empirical root distribution data of white alder (Alnus incana L. [

69]) was used because this is the only riparian species for which the root distribution model was calibrated. DBH of the four trees were 8.5 cm, 10.0 cm, 7.5 cm and 8.0 cm, respectively. Fine root density

${D}_{roots}$ (m m

^{−2}) is then calculated as:

where fine roots have diameters smaller than 1.5 mm,

${N}_{roots\_tot}$ are the total number of fine roots per diameter class using the pipe theory approach,

${d}_{stem}$ is the horizontal distance (m), or elongated position from the tree stem at which root density is calculated for, and

${d}_{stemmax}$ is the maximum lateral extent of the root system (m). The estimation of root frequencies with diameters greater than 1.5 mm is done by using a gamma function as presented in Schwarz et al. [

67]. The sum of the roots’ surface area per diameter class at position

${d}_{stem}$ is then divided by the area plot.