I have a raster grid and some shapes (see figure below, in which only one of the shapes red is depicted). For each raster cell, I am checking how much of the area of the raster cell is overlapping with any of the shapes. Afterwards, I am multiplying the overlapping areas with a value that represents a property of the raster cell (e.g. population) and sum this up (equivalent to Microsoft Excel's SUMPRODUCT() function).
How would I write a corresponding equation in mathematical notation?
Raster cell index i,j
Shape file index k
If $\{S_i\}_{i=1}^{m}$ is a collection of shapes, and $\{ R_j \}_{j=1}^{n}$ is a collection of raster cells, then for each shape $S_i$ the 'sumproduct' would be:
$$ \mathrm{sp}(S_i) = \sum_{j = 1}^{n} {| R_j \cap S_i |} $$
where the $| \cdot |$ denotes area and the $\cap$ denotes intersection. If the shape is contained entirely 'inside' the raster cells, you would have $\mathrm{sp}(S_i) = |S_i|$.
If you don't care which shape the raster cell intersects, and you don't double count when it intersects more than once, then let ${\bf S} = \cup_{i=1}^{m} S_i$ be the union of all the shapes. What you want in this case is:
$$ \mathrm{sp_\mathrm{tot}} = \sum_{j = 1}^{n} {| R_j \cap {\bf S} |} \mbox{.} $$
If you do want to double count, it is instead the double sum:
$$ \mathrm{sp_\mathrm{dbl}} =\sum_{i=1}^{m}\sum_{j=1}^{n} | R_j \cap S_i | = \sum_{i=1}^{m} \mathrm{sp}(S_i) \mbox{.} $$