# Damage functions at the elementary component level

For a given combination of water depth and flood duration, we define the damage to an elementary component as the expected value of the cost of the actions that are necessary to go back from the state induced by the flood to the normal state. Thus, for a given elementary component the damage resulting from a flood is given by the following equation:

\[ d_e(h, d) = \sum\limits_{s \in S} p_s(h, d) \times i_s(h, d) \times c_{a_s} \times h_{a_s}(h, d) \tag{1}\]

where:

- \(e\): refers to the elementary component
- \(h\): refers to the water depth relative to \(e\). \(h\) may be negative, to take into account impact of humidity.
- \(d\): refers to the duration of submersion relative to \(e\). \(d\) has to be strictly positive for a submersion to occur.
- \(S\): refers to the set of all admissible states \(s\) for \(e\).
- \(p_s(h, d)\): refers to the probability that a submersion of characteristics \(h\) and \(d\) brings \(e\) into the state \(s\).
- \(i_s(h, d)\): refers to the intensity in which a submersion of characteristics \(h\) and \(d\) brings \(e\) into the state \(s\).
- \(c_{a_{s}}\): refers to the cost of the action \(a\) that is necessary to bring back \(e\) from the state \(s\) to the normal one.
- \(h_{a_{s}}(h, d)\): refers to the height up to which it is necessary to perform the action \(a\) for a submersion of characteristics \(h\) and \(d\). If \(e\) is indivisible, \(h_a(h, d)\) is constant and equal to the height of the elementary component.

Examples are given in this section to illustrate the elements of equation Equation 1.