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I create channel network and catchment area from a DEM with SAGA GIS. I know that flow is starting from a cell to an other cell. Firstly, a cell is filled with water and then it is starting to flow into an other cell.

"Flow accumulation" create accumulated flow into each cell. It is about number of cells.

But I want to know, How can I determine how much water accumulate in cells? Do I need a volume layer to solve this?

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In spatial hydrology, DEM-based flow accumulation operations are typically static. That is, they represent a steady-state condition of the discharge of surface and near surface water passing through a point. Flow accumulation grids are actually accumulating contributing area downslope, i.e. areas not volumes. The accumulated cells that you described is because clearly the number of upslope pixels is related to the upslope area through the grid cell area. However, in addition to total number of upslope grid cells, it is common to represent this value as a 'specific contributing area' which is normalized for the contour length through the cell. Regardless of the units, the idea is that the contributing area can be used as a surrogate for the discharge, and for most locations this assumption is quite good in that there is a strong (often non-linear) relation between upslope area and discharge (at least for most streams).

However, you're talking about accumulating water volume and that requires a bit more information and likely implies that you would need to consider the dynamics of a full rainfall-runoff model. In particular, in addition to the upslope inputs (which the flow accumulation operation represent), you'd also need to account for the various losses in each grid cell due to evapotranspiration, percolation to deep groundwater, etc. This will be a function of much more than simply the topography, such as vegetation type and density, soil properties, the relation with the ground water table and much more. You may even need to account for spatial variation in the input of rainfall. Rainfall-runoff models usually require substantial parameterization to account for these factors. Thus, the topographically driven flow routing component represented by a flow accumulation grid would only be one such parameter (accounting for the redistribution of water within catchments) in a dynamic model which would then allow you to compute flow volumes. For an example of such a model, TOPMODEL is a good and easy-to-conceptualize starting point.

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