An integrated approach for evaluating the effectiveness of landslide risk reduction in unplanned communities in the Caribbean

Abstract

Despite the recognition of the need for mitigation approaches to landslide risk in developing countries, the delivery of ‘on-the-ground’ measures is rarely undertaken. With respect to other ‘natural’ hazards, it is widely reported that mitigation can pay. However, the lack of such an evidence base in relation to landslides in developing countries hinders advocacy amongst decision makers for expenditure on ex-ante measures. This research addresses these limitations directly by developing and applying an integrated risk assessment and cost–benefit analysis of physical landslide mitigation measures implemented in an unplanned community in the Eastern Caribbean. In order to quantify the level of landslide risk reduction achieved, landslide hazard and vulnerability were modelled (before and after the intervention), and project costs, direct and indirect benefits were monetised. It is shown that the probability of landslide occurrence has been substantially reduced by implementing surface-water drainage measures and that the benefits of the project outweigh the costs by a ratio of 2.7–1. This paper adds to the evidence base that ‘mitigation pays’ with respect to landslide risk in the most vulnerable communities—thus strengthening the argument for ex-ante measures. This integrated project evaluation methodology should be suitable for adoption as part of the community-based landslide mitigation project cycle, and it is hoped that this resource, and the results of this study, will stimulate further such programmes.

Appendix: Deriving the present value of the expected costs of landslides of different types

1.1 Probabilities

Let:

• p _A,t  = probability of a landslide of type A occurring in year t

• p _B,L,t  = probability of a landslide of type B occurring in location L in year t

• q _A  = 0.1

• q _B  = 0.2

• q _A,I  = 0.01 if intervention has occurred, 0.1 otherwise

• q _B,I  = 0.02 if intervention has occurred, 0.2 otherwise

• N = project lifetime

Assume:

$$ p_{A,t} = \left\{ {\begin{array}{*{20}c} {q_{A,I} } \hfill & {{\text{if}}\quad 0 \le t \le N - 1,\,{\text{and no landslide of type }}A\,{\text{in year }}i,{\text{for all }}i < t} \hfill \\ {q_{A} } \hfill & {{\text{if}}\quad t \ge N,\,{\text{and no landslide of type }}A\,{\text{in year }}i,\,{\text{for all }}i < t} \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. $$

$$ p_{B,L,t} = \left\{ {\begin{array}{*{20}c} {\left( { 1- q_{A,I} } \right)q_{B,I} } \hfill & {{\text{if }}0 \le t \le N - 1,\,{\text{and no landslide of type }}A\,{\text{in year}}\,i,\,{\text{and no landslide of type }}B\,{\text{in location }}L\,{\text{in year}}\,i,\,{\text{for all}}\,i < t} \hfill \\ {\left( { 1-\, q_{A} } \right)q_{B} } \hfill & {{\text{if }}t \ge N,\,{\text{and no landslide of type }}A\,{\text{in year}}\,i,\,{\text{and no landslide of type }}B\,{\text{in location }}L\,{\text{in year }}i,\,{\text{for all }}i < t} \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. $$

That is, a landslide of type A may occur at most once—once it has occurred, the landscape is sufficiently altered that it may not occur again. The same goes for a landslide of type B: there are three locations in which this type of landslide may occur, and once it has occurred in a particular location, it may not occur in that location again. An additional restriction is that a landslide of type A affects the entire area in which landslides of type B may occur: once a landslide of type A has occurred, no landslide of type B may occur in any location. It is further assumed that if a landslide of type A occurs in a given year, no landslide of type B occurs that year.

Then prior probabilities are:

$$ p_{A,t} = \left\{ {\begin{array}{*{20}c} {\left( { 1- q_{A,I} } \right)^{t} q_{A,I} } \hfill & {{\text{for all }}0 \le t \le N - 1} \hfill \\ {\left( { 1- q_{A,I} } \right)^{N} \left( { 1- q_{A} } \right)^{t - N} q_{A} } \hfill & {{\text{for all }}t \ge N} \hfill \\ \end{array} } \right. $$

$$ p_{B,L,t} = \left\{ {\begin{array}{*{20}c} {\left( { 1- q_{A,I} } \right)^{t} \left( { 1- q_{B,I} } \right)^{t} \left( { 1- q_{A,I} } \right)q_{B,I} } \hfill & {{\text{for all }}0 \le t \le N - 1} \hfill \\ {\left( { 1- q_{A,I} } \right)^{N} \left( { 1- q_{B,I} } \right)^{N} \left( { 1- q_{A} } \right)^{t - N} \left( { 1- q_{B} } \right)^{t - N} \left( { 1- q_{A} } \right)q_{B} } \hfill & {{\text{for all }}t \ge N} \hfill \\ \end{array} } \right. $$

1.2 Present values

Let:

• r = discount rate (constant)

• δ _t  = discount factor for year t = 1/(1 + r) ^t

• c _A  = Cost of a landslide of type A occurring

• c _B,L  = Cost of a landslide of type B occurring in location L

• PV _A,y  = Present value of expected costs from landslides of type A occurring in year y

• PV _B,L,y  = PV of expected costs from landslides of type B occurring in location L in year y

Then:

$$ {\text{PV}}_{A,t} = \delta_{t} p_{A,t} c_{A} = p_{A,t} c_{A} /( 1 + r)^{t} $$

And:

$$ {\text{PV}}_{B,L,t} = \delta_{t} p_{B,L,t} c_{B,L} = p_{B,L,t} c_{B,L} /( 1 + r)^{t} $$

So the present value of expected costs from landslides of type A is:

$$ \begin{aligned} & \left[ {{\text{ PV}}_{A,0} + {\text{PV}}_{A, 1} + \cdots + {\text{PV}}_{A,N - 1} } \right] + \left[ {{\text{ PV}}_{A,N} + {\text{PV}}_{A,N + 1} + \ldots } \right] \\ = & \left[ {p_{A,0} c_{A} + \delta_{ 1} p_{A, 1} c_{A} + \cdots + \delta_{N - 1} p_{A,N - 1} c_{A} } \right] + \left[ {\delta_{N} p_{A,N} c_{A} + \delta_{N + 1} p_{A,N + 1} c_{A} + \ldots } \right] \\ = & \left[ {q_{A,I} c_{A} + q_{A,I} c_{A} \left( { 1- q_{A,I} } \right)/( 1 + r) + \cdots + q_{A,I} c_{A} \left( { 1- q_{A,I} } \right)^{N - 1} /\left( { 1 + r} \right)^{N - 1} } \right] + \left[ {q_{A} c_{A} \left( { 1- q_{A,I} } \right)^{N} /\left( { 1 + r} \right)^{N} + q_{A} c_{A} \left( { 1- q_{A} } \right)\left( { 1- q_{A,I} } \right)^{N} /( 1 + r)^{N + 1} + \cdots } \right] \\ \end{aligned} $$

which is a pair of geometric series, and so:

$$ = \frac{{q_{A,I} c_{A} \left( {1 - \left[ {{{\left( {1 - q_{A,I} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - q_{A,I} } \right)} {\left( {1 + r} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + r} \right)}}} \right]^{N} } \right)}}{{1 - \left[ {{{\left( {1 - q_{A,I} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - q_{A,I} } \right)} {\left( {1 + r} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + r} \right)}}} \right]}} + \frac{{q_{A} c_{A} \left[ {{{\left( {1 - q_{A,I} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - q_{A,I} } \right)} {\left( {1 + r} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + r} \right)}}} \right]^{N} }}{{1 - \left[ {{{\left( {1 - q_{A} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - q_{A} } \right)} {\left( {1 + r} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + r} \right)}}} \right]}} $$

And the present value of expected costs from landslides of type B in location L is:

$$ \begin{aligned} & \left[ {{\text{PV}}_{B,L,0} + {\text{PV}}_{B,L, 1} + \cdots + {\text{PV}}_{B,L,N - 1} } \right] \\ &+ \left[ {{\text{PV}}_{B,L,N} + {\text{PV}}_{B,L,N + 1} + \ldots } \right] \hfill \\ & = \left[ {p_{B,L,0} c_{B,L} + \delta_{ 1} p_{B,L, 1} c_{B,L} + \cdots + \delta_{N - 1} p_{B,L,N - 1} c_{B,L} } \right] \\ &+ \left[ {\delta_{N} p_{B,L,N} c_{B,L} + \delta_{N + 1} p_{B,L,N + 1} c_{B,L} + \ldots } \right] \hfill \\ &= [q_{B,I} c_{B,L} \left( { 1- q_{A,I} } \right) + {{q_{B,I} c_{B,L} \left( { 1- q_{A,I} } \right)\left( { 1- q_{A,I} } \right)\left( { 1- q_{B,I} } \right)} \mathord{\left/ {\vphantom {{q_{B,I} c_{B,L} \left( { 1- q_{A,I} } \right)\left( { 1- q_{A,I} } \right)\left( { 1- q_{B,I} } \right)} {\left( { 1 + r} \right)}}} \right. \kern-\nulldelimiterspace} {\left( { 1 + r} \right)}} + \cdots + {{q_{B,I} c_{B,L} \left( { 1- q_{A,I} } \right)\left( { 1- q_{A,I} } \right)^{N - 1} \left( { 1- q_{B,I} } \right)^{N - 1} } \mathord{\left/ {\vphantom {{q_{B,I} c_{B,L} \left( { 1- q_{A,I} } \right)\left( { 1- q_{A,I} } \right)^{N - 1} \left( { 1- q_{B,I} } \right)^{N - 1} } {\left( { 1 + r)^{N - 1} } \right]}}} \right. \kern-\nulldelimiterspace} {\left( { 1 + r)^{N - 1} } \right]}} \hfill \\ &+\left[ {{{q_{B} c_{B,L} \left( { 1- q_{A} } \right)\left( { 1- q_{A,I} } \right)^{N} \left( { 1- q_{B,I} } \right)^{N} } \mathord{\left/ {\vphantom {{q_{B} c_{B,L} \left( { 1- q_{A} } \right)\left( { 1- q_{A,I} } \right)^{N} \left( { 1- q_{B,I} } \right)^{N} } {\left( { 1 + r} \right)^{N} }}} \right. \kern-\nulldelimiterspace} {\left( { 1 + r} \right)^{N} }} + {{q_{B} c_{B,L} \left( { 1- q_{A} } \right)\left( { 1- q_{A,I} } \right)^{N} \left( { 1- q_{B,I} } \right)^{N} \left( { 1- q_{A} } \right)\left( { 1- q_{B} } \right)} \mathord{\left/ {\vphantom {{q_{B} c_{B,L} \left( { 1- q_{A} } \right)\left( { 1- q_{A,I} } \right)^{N} \left( { 1- q_{B,I} } \right)^{N} \left( { 1- q_{A} } \right)\left( { 1- q_{B} } \right)} {\left( { 1 + r} \right)^{N + 1} }}} \right. \kern-\nulldelimiterspace} {\left( { 1 + r} \right)^{N + 1} }} + \ldots } \right] \hfill \\ \end{aligned} $$

which is a pair of geometric series, and so:

$$ = \frac{{q_{B,I} c_{B,L} \left( {1 - q_{A,I} } \right)\left( {1 - \left[ {{{\left( {1 - q_{A,I} } \right)\left( {1 - q_{B,I} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - q_{A,I} } \right)\left( {1 - q_{B,I} } \right)} {\left( {1 + r} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + r} \right)}}} \right]^{N} } \right)}}{{1 - \left[ {{{\left( {1 - q_{A,I} } \right)\left( {1 - q_{B,I} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - q_{A,I} } \right)\left( {1 - q_{B,I} } \right)} {\left( {1 + r} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + r} \right)}}} \right]}} + \frac{{q_{B} c_{B,L} \left( {1 - q_{A} } \right)\left[ {{{\left( {1 - q_{A,I} } \right)\left( {1 - q_{B,I} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - q_{A,I} } \right)\left( {1 - q_{B,I} } \right)} {\left( {1 + r} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + r} \right)}}} \right]^{N} }}{{1 - \left[ {{{\left( {1 - q_{A} } \right)\left( {1 - q_{B} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - q_{A} } \right)\left( {1 - q_{B} } \right)} {\left( {1 + r} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + r} \right)}}} \right]}} $$