The surface storage is not treated as a sharp threshold: runoff takes place before the water level reaches some threshold value of average surface storage. This is done because runoff at a micro-scale is a spatial process of ponds that fill up and overflow into each other. The gridcell in LISEM represents an area of for instance 10×10 m2 and before the average storage of this surface is filled the water at the edges of this area is moving downstream. The average depression storage can be measured at a given scale. The start value is more difficult and in LISEM a pragmatic approach is taken, based on the fraction of the surface that is ponded.
Surface storage is calculated using the Maximum Depression Storage (MDS). This is the threshold value of a given area above which surface micro depressions overflow: all depressions are connected to each other and to an point outside the reference area so that in principle each additional “drop” of water will flow outside the reference area. The MDS is determined by Kamphorst et al. (2000) from 221 digital elevation models of various types of micro relief, in a wide variety of agricultural circumstances and soil types. The analysis is based on DEMs of roughly 1m2. Figure 1 shows the relation between the MDS and Random Roughness (RR). Originally the depression storage in LISEM was based on the work of Onstad (1984). Using the same form of equation Kamphorst et al. (2000) found:
MDS = 0.243RR + 0.010RR2 + 0.012RRS R2=0.88, n=221
in which RR is the standard deviation of surface heights (cm) and S is the terrain slope (%). They tested 6 different roughness indices but found that the standard deviation of the heights gave the best relation with MDS (cm).
Figure 1. Maximum depression storage as a function of Random Roughness
(after Kamphorst et al.,2000)
Apart from the water available for runoff, the roughness also determines width of the overland flow in LISEM. Rather then taking the cell width dx, the flow width (and hydraulic radius) is assumed linearly related to the fraction of ponded surface fpa in the cell. The latter variable is related to the water depth at the surface h (mm) (Jetten and De Roo, 2001):
fpa = 1-exp(-ah)
where a is an empirical factor between 0.04 and 1.8 for the roughness data set mentioned above. Figure 2 shows that the factor a appears to be strongly related to RR (in mm):
a = 1.406*(RR)-0.942 R 2=0.99, n=362
In which RR is in mm. Figure 3 shows the increase of ponded area with water height. Although not all the surfaces had a random roughness but showed also oriented roughness elements, the RR appeared to be the best roughness index to explain the variance in the dataset. Tortuosity based indices performed less as did the Linden -Van Dooren indices (Kamphorst et al., 2000).
Figure 2. Ponded area shape factor a related to RR (Jetten and de Roo, 2001).
Figure 3. Ponded area fraction increase with water height for
different values of RR (increase with 5 mm intervals).
Based on the MDS value and the fraction of ponded area the runoff before the water level reaches the MDS height is calculated as follows. It is assumed that the runoff starts if 10% of the surface is ponded so that the equation above for fpa reads:
0.1 = 1-exp(-a h)
which means that at h = ln(0.9)/-a, runoff starts. If this threshold for h, called here Start Depressional Storage or SDS is larger than MDS, 0.9 of MDS is taken. Between this starting value “SDS” and the MDS value the runoff gradually increases in a nonlinear way to reflect the observation that the surface usually consists of degraded aggregates which “release” little runoff until they are fully submerged. After the water level has passed MDS the runoff height increases linearly with water height:
h < SDS : runoff = 0
h > SDS : runoff = (h-SDS) * (1-exp(-h*(h-SDS)/(MDS-SDS)))
In fact this resembles a Gaussian curve (1-exp(-x2)) as is seen in figure 4.
Figure 4. Increase of runoff height with with water height for
different values of MDS and SDS (increase with 5 mm intervals).