Landslide Susceptibility along the Sea to Sky Highway


	Theory We used the simplest, physically-based model that would capture the topographic effects relevant to landslide susceptibility on hillslopes. As shown below, Montgomery and Dietrich's (1994) model was simplified, with many parameters requiring expert knowledge for assumed values. The value of such a model is that: It can be applied in diverse environments without costly attempts at parameterization; Results from different sites can be directly compared; and It takes little special training to use the model (Dietrich and Montgomery, 1998). Essentially, the model is a slope stability model coupled with a hydrological model, which solves for the critical rainfall needed to trigger slope instability. Parameters referred to in following equations are summarized in Table 1 below. Table 1 - Parameters *Slope Stability Model* The model is used to find the critical steady state rainfall (Qc) necessary to trigger slope instability. It is based on an infinite slope model form of the Mohr-Coulomb failure law in which the downslope component of the weight of the soil just at failure, is equal to the strength of resistance caused by cohesion (soil cohesion and/or root strength), (C) and by frictional resistance due to the effective normal stress on the failure plane: [1] A further simplification is to set the cohesion to zero, due to the assumption of the soil being fully saturated. Dietrich and Montgomery (1998) state that this approximation is clearly incorrect in most applications, but the rocky, sandy soils of colluvial mantled landscapes probably have minor soil cohesion in [1]. The model neglects the role of root cohesion for several important reasons; the first being that root strength varies widely, both spatially and temporally. For watershed scale modeling, parameterization of root strength patterns across the landscape would be very time consuming and very expensive to collect the data (Dietrich and Montgomery, 1998). The lack of root cohesion is compensated for by setting the friction angle to a high, but acceptable value. We varied the friction angle from 40 degrees to 50 degrees to see the effects of varying the friction angle on slope stability. By eliminating cohesion, [1] can be written as: [2] According to Dietrich and Montgomery (1998) equation [2] states that the soil does not necessarily have to be saturated for failure. While this is nearly always assumed when one analyses a landslide scar, theoretically it is not necessary. When the slope angle is greater than or equal to the friction angle then the slope is steep enough to fail whether or not it is saturated, these slopes are categorised as unconditionally unstable. *Hydrological Model* Dietrich and Montgomery (1998) identified the hydrologic controls of a steady state shallow subsurface flow, to model the amount of saturation of the soil. They assumed that the steady state hydrologic response model mimics what the relative spatial pattern of wetness would be during an intense natural storm and the routing of water off the landscape. It was assumed that there is no overland flow, no significant deep drainage, and no significant flow in the bedrock, then q, the effective precipitation (rainfall minus evapotranspiration) times the upslope drainage area, 'a', must be the amount of runoff that occurs through a particular grid cell of width 'b' under steady state conditions. The model is based on Darcy's law and was rewritten; [3] where sin(theta) is the head gradient. At saturation, the shallow subsurface flow will equal the transmissivity, T, (the vertical integral of the saturated conductivity) times the head gradient, sin theta and the width of the outflow boundary, b and this we can approximate as follows: [4] Combining [3] and [4] leads to: [5] Equation [5] is a hydrologic ratio and a topographic ratio. The hydrologic ratio is q/T. This ratio captures the magnitude of the precipitation event, represented by q, relative to the subsurface ability to convey the water downslope, i.e. the transmissivity (Dietrich and Montgomery, 1998). It can be seen that the larger the q relative to T, the more likely the ground will be completely saturated, thus the more area on the hillslope that will become unstable. The topographic ratio, a / (b * sin(theta)), essentially captures the effects of topographic control and the runoff. The steeper the slopes, the faster the subsurface flow and the consequently the lower the relative wetness defined by h/z (Dietrich and Montgomery, 1998). The actual effective steady state rainfall can be calculated if the transmissivity is estimated to be 65 m2/day, this value was obtained from Dietrich and Montgomery (1998). The model assumes a steady state rainfall; this gives a good indication of the effects of brief rainstorms whose short term effective rainfall is greater than the steady state value. *Coupled Hydrological and Slope Stability Model* By combining the slope stability model [2] and the hydrologic model [5] we could solve for either the hydrologic ratio: [6a] or the area per outflow boundary length [6b] Equation [6] is the coupled hydrologic-slope stability equation. The model has three topographic terms that are defined by the digital elevation model (DEM): drainage area, a, outflow boundary length, b, and hillslope angle, theta. Using the equation [6b], Dietrich and Montgomery (1998), indicate that by rearranging the equation one can solve for different areas using stability fields, as well as the critical steady state rainfall required for slope instability. These fields are unconditionally stable, unconditionally unstable, stable, and unstable respectively, as shown in Figure 1 below. The model used to calculate the critical steady state rainfall is shown in Figure 2. Figure 1- Slope Stability Conditions Figure 2 - Slope Stability Model Back to the Top Home

Theory

We used the simplest, physically-based model that would capture the topographic effects relevant to landslide susceptibility on hillslopes. As shown below, Montgomery and Dietrich's (1994) model was simplified, with many parameters requiring expert knowledge for assumed values. The value of such a model is that:

It can be applied in diverse environments without costly attempts at parameterization;
Results from different sites can be directly compared; and
It takes little special training to use the model (Dietrich and Montgomery, 1998).

Essentially, the model is a slope stability model coupled with a hydrological model, which solves for the critical rainfall needed to trigger slope instability.
Parameters referred to in following equations are summarized in Table 1 below.

Table 1 - Parameters

Slope Stability Model

The model is used to find the critical steady state rainfall (Qc) necessary to trigger slope instability. It is based on an infinite slope model form of the Mohr-Coulomb failure law in which the downslope component of the weight of the soil just at failure, is equal to the strength of resistance caused by cohesion (soil cohesion and/or root strength), (C) and by frictional resistance due to the effective normal stress on the failure plane:

[1]

A further simplification is to set the cohesion to zero, due to the assumption of the soil being fully saturated. Dietrich and Montgomery (1998) state that this approximation is clearly incorrect in most applications, but the rocky, sandy soils of colluvial mantled landscapes probably have minor soil cohesion in [1].
The model neglects the role of root cohesion for several important reasons; the first being that root strength varies widely, both spatially and temporally. For watershed scale modeling, parameterization of root strength patterns across the landscape would be very time consuming and very expensive to collect the data (Dietrich and Montgomery, 1998).

The lack of root cohesion is compensated for by setting the friction angle to a high, but acceptable value. We varied the friction angle from 40 degrees to 50 degrees to see the effects of varying the friction angle on slope stability. By eliminating cohesion, [1] can be written as:

[2]

According to Dietrich and Montgomery (1998) equation [2] states that the soil does not necessarily have to be saturated for failure. While this is nearly always assumed when one analyses a landslide scar, theoretically it is not necessary. When the slope angle is greater than or equal to the friction angle then the slope is steep enough to fail whether or not it is saturated, these slopes are categorised as unconditionally unstable.

Hydrological Model

Dietrich and Montgomery (1998) identified the hydrologic controls of a steady state shallow subsurface flow, to model the amount of saturation of the soil. They assumed that the steady state hydrologic response model mimics what the relative spatial pattern of wetness would be during an intense natural storm and the routing of water off the landscape. It was assumed that there is no overland flow, no significant deep drainage, and no significant flow in the bedrock, then q, the effective precipitation (rainfall minus evapotranspiration) times the upslope drainage area, 'a', must be the amount of runoff that occurs through a particular grid cell of width 'b' under steady state conditions. The model is based on Darcy's law and was rewritten;

[3]

where sin(theta) is the head gradient. At saturation, the shallow subsurface flow will equal the transmissivity, T, (the vertical integral of the saturated conductivity) times the head gradient, sin theta and the width of the outflow boundary, b and this we can approximate as follows:

[4]

Combining [3] and [4] leads to:

[5]

Equation [5] is a hydrologic ratio and a topographic ratio. The hydrologic ratio is q/T. This ratio captures the magnitude of the precipitation event, represented by q, relative to the subsurface ability to convey the water downslope, i.e. the transmissivity (Dietrich and Montgomery, 1998). It can be seen that the larger the q relative to T, the more likely the ground will be completely saturated, thus the more area on the hillslope that will become unstable. The topographic ratio, a / (b * sin(theta)), essentially captures the effects of topographic control and the runoff. The steeper the slopes, the faster the subsurface flow and the consequently the lower the relative wetness defined by h/z (Dietrich and Montgomery, 1998).
The actual effective steady state rainfall can be calculated if the transmissivity is estimated to be 65 m2/day, this value was obtained from Dietrich and Montgomery (1998). The model assumes a steady state rainfall; this gives a good indication of the effects of brief rainstorms whose short term effective rainfall is greater than the steady state value.

Coupled Hydrological and Slope Stability Model

By combining the slope stability model [2] and the hydrologic model [5] we could solve for either the hydrologic ratio:

[6a]

or the area per outflow boundary length

[6b]

Equation [6] is the coupled hydrologic-slope stability equation. The model has three topographic terms that are defined by the digital elevation model (DEM): drainage area, a, outflow boundary length, b, and hillslope angle, theta.
Using the equation [6b], Dietrich and Montgomery (1998), indicate that by rearranging the equation one can solve for different areas using stability fields, as well as the critical steady state rainfall required for slope instability. These fields are unconditionally stable, unconditionally unstable, stable, and unstable respectively, as shown in Figure 1 below. The model used to calculate the critical steady state rainfall is shown in Figure 2.

Figure 1- Slope Stability Conditions

Figure 2 - Slope Stability Model

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