Runoff production and runoff attenuation index
For each pixel \(i\), defined by a land use type and soil characteristics, we estimate runoff \(Q\) (mm) with the Curve Number method:
(1)\[\begin{split}Q_{p,i} = \begin{Bmatrix} \frac{(P  \lambda S_{max_i})^2}{P + (1\lambda) S_{max,i}} & if & P > \lambda \cdot S_{max,i} \\ 0 & & otherwise \end{Bmatrix}\end{split}\]
Where \(P\) is the design storm depth in mm, \(S_{max,i}\) is the potential retention in mm, and \(\lambda \cdot S_{max}\) is the rainfall depth needed to initiate runoff, also called the initial abstraction (\(\lambda=0.2\) for simplification).
\(S_{max}\) (calculated in mm) is a function of the curve number, \(CN\), an empirical parameter that depends on land use and soil characteristics (NRCS 2004):
(2)\[S_{max,i}=\frac{25400}{CN_i}254\]
The model then calculates runoff retention per pixel \(R_i\) as:
(3)\[R_i=1\frac{Q_{p,i}}{P}\]
And runoff retention volume per pixel \(R\_m3_i\) as:
(4)\[R\_m3_i=R_i\cdot P\cdot pixel.area\cdot 10^{3}\]
With \(pixel.area\) in \(m^2\).
Runoff volume (also referred to as “flood volume”) per pixel \(Q\_m3_i\) is also calculated as:
(5)\[Q\_m3_i=Q_{p,i}\cdot P\cdot pixel.area\cdot 10^{3}\]
Calculate potential service (optional)
The service is the monetary valuation of avoided damage to built infrastructure and number of people at risk. As of this version of InVEST, the population metrics described here are not yet implemented.
For each watershed (or sewershed) with floodprone areas, compute:

Affected.Pop : total potential number of people affected by flooding (could focus on vulnerable groups only, e.g. related to age, language, etc. See Arkema et al., 2017, for a review of social vulnerability metrics). This metric is calculated by summing the population in the intersection of the two shapefiles (watershed and floodprone area).

\(Affected.Build\) : sum of potential damage to built infrastructure in $, This metric is calculated by multiplying building footprint area within the watershed and potential damage values in \(m^2\).
Aggregation of runoff retention and potential service values at the watershed scale
For each watershed, compute the following indicator of the runoff retention service:
(6)\[Service.built=Affected.Build\sum_{watershed}0.001(PQ_{p,i})\cdot pixel.area\]
where \(pixel.area\) is the pixel area (\(m^2\)), \(Service.built\) is expressed in \(m^3\).
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