If your data is in raster format you can correct the individual rasters (A, B, C and D) to have a value 0 in case of NoData. Then just sum the new rasters together and perform a zonal statistics (SUM) of the raster using your polygons as zones layer.
If the value grids are polygons (vector layers), you could merge the layers together to create featureclass with regions (overlapping polygons). When you do an intersect between your zones (also polygons) and the regions, it will intersect the regions themselves too and create smaller overlapping polygons. The stacked polygons can be use to dissolve the features and apply a Sum while dissolving. For this you will need a new key column, which you could create from the zones ID and the Shape_Area. See image below with the intersect and dissolve results:
If your data is in raster format you can correct the individual rasters (A, B, C and D) to have a value 0 in case of NoData. Then just sum the new rasters together and perform a zonal statistics (SUM) of the raster using your polygons as zones layer.
If the value grids are polygons (vector layers), you could merge the layers together to create featureclass with regions (overlapping polygons). When you do an intersect between your zones (also polygons) and the regions, it will intersect the regions themselves too and create smaller overlapping polygons. The stacked polygons can be use to dissolve the features and apply a Sum while dissolving. For this you will need a new key column, which you could create from the zones ID and the Shape_Area. See image below with the intersect and dissolve results:
