Erosion/deposition modeling with USPED

UNDER PREPARATION

Helena Mitasova, Lubos Mitas, University of Illinois at Urbana-Champaign

Copyright © 1999 Helena Mitasova and Lubos Mitas
(note: this document contains unpublished material, please cite this document and a paper Mitasova et al 1996 when using this material in publications)

1. Method

USPED (Unit Stream Power - based Erosion Deposition) is a simple model which predicts the spatial distribution of erosion and deposition rates for a steady state overland flow with uniform rainfall excess conditions for transport capacity limited case of erosion process. For this case, we assume that the critical shear stress is negligible and the sediment flow rate qs(r) is at the sediment transport capacity T(r), r=(x,y) which is approximated by (Julien and Simons 1985)
 
|qs(r)|  =  T(r) =  Kt (r) |q(r)|m sin b(r)n
 
 

where b(r) is slope,  q(r) is water flow rate, Kt(r) is transportability coefficient dependent on soil and cover, m, n are constants depending on the type of flow and soil properties. For overland flow the constants are usually set to m=1.6, n=1.3 (Foster 1993). Water flow can be expressed as a function of upslope contributing area  A(r)

|q(r)| =  A(r) i

where i[m] is uniform rainfall intensity (note: approximation by upslope area neglects the change in flow velocity due to cover). For the uniform soil and cover properties represented by Kt, the  net erosion/deposition rate is estimated as a divergence of the sediment flow (see Appendix in Mitas and Mitasova 1998):

ED(r) = div qs(r) = Kt { [grad h(r)] . s(r) sin b(r) - h(r) [kp(r) + kt(r)] }

where s(r) is the unit vector in the steepest slope direction, h(r) is the water depth estimated from the upslope area A(r), kp(r) is the profile curvature (terrain curvature in the direction of the steepest slope), kt(r) is the tangential curvature (curvature in the direction tangential to a contourline projected to the normal plane). Topographic parameters s(r), kp(r), kt(r) are computed from the first and second order derivatives of terrain surface approximated e.g., by the RST (Mitasova and Mitas, 1993; Mitasova and Hofierka, 1993; Krcho 1991). According to the 2D formulation, the spatial distribution of erosion and deposition is controlled by the change in the overland flow depth (first term) and by the local geometry of terrain (second term), including both profile and tangential curvatures. The bivariate formulation thus demonstrates that the local acceleration of flow in both the gradient and tangential directions (related to the profile and tangential curvatures) play equally important roles in spatial distribution of erosion/deposition. The interplay between the magnitude of water flow change and both terrain curvatures reflected in the bivariate formulation determines whether erosion or deposition will occur.

No experimental work was performed to develop parameters needed for USPED, therefore we
use the USLE or RUSLE parameters to incorporate the impact of soil and cover and obtain at least a relative estimate of net erosion and deposition.We assume that we can estimate sediment flow at sediment transport capacity as

T = R K LS C P

where R=im, KCP=Kt and LS=Am sin b n (add units here), and m=1.6, n=1.3 for prevailing rill erosion while m=n=1 for prevailing sheet erosion. Then the net erosion/deposition is estimated as

ED  =  div (T . s)  =  d(T*cos a)/dx  +  d(T*sin a)/dy

where a [deg] is aspect of the terrain surface.This equation is equivalent to the relationship with curvatures presented above, however the computational procedure is simpler.
There are no experiments or theory available to show that KCP is a linear function of Kt so the results should be used with caution.Soil and cover parameters similar to those used in USLE or WEPP were not developed for the USPED model, because no systematic experimental work was performed. However, it was suggested (Moore and Wilson 1992) that under certain conditions, there is a relation between the USPED concept and USLE. It is therefore possible to combine the model with the USLE/RUSLE factors to predict the relative impact of land use change on erosion/deposition pattern assuming the conditions valid for USPED hold.  Caution should be used when interpreting the results because the USLE parameters were developed for simple plane fields and to obtain accurate predictions for complex terrain conditions they need to be re-calibrated ( Foster 1990, Mitasova et al 1997 reply).
 

2. GIS implementation

2.1  GRASS GIS

Given data: 

raster: elevation, K, C, (P)
constant: R
Computation

Copyright © 1999 Helena Mitasova and Lubos Mitas 

1. r.flow elevation dsout=flowacc
2. r.slope.aspect elevation slope=slope aspect=aspect
3. r.mapcalc
      lsfac=exp(flowacc*resolution,1.6)*exp(sin(slope),1.3)
  or  lsfac=flowacc*resolution*sin(slope)
      qsx = R * K * C * lsfac * cos(aspect)
      qsy = R * K * C * lsfac * sin(aspect)
4. r.slope.aspect qsx dx = qsx.dx
5. r.slope.aspect qsy dy = qsy.dy
6. r.mapcalc
        erdep = qsx.dx + qsy.dy
2.2 ArcView-Spatial Analyst

Given data 

grid: elevation, K, C, (P)
constants: R
Computation

Copyright © 1999 Helena Mitasova and Lubos Mitas 

1. FlowDirection elevation flowdir
   FlowAccumulation flowdir flowacc
2. DERIVE SLOPE elevation slope
3. DERIVE ASPECT elevation aspect
4. MAP CALCULATOR
   lsfac=Pow(flowacc*resolution,1.6)*Pow(Sin(slope),1.3)
  or  lsfac=flowacc*resolution*Sin(slope)
       qsx = R * K * C * lsfac * Cos(aspect)
       qsy = R * K * C * lsfac * Sin(aspect)
5. DERIVE SLOPE  qsx qsx.slope
6. DERIVE ASPECT qsx qsx.aspect
7. DERIVE SLOPE qsy qsy.slope
8. DERIVE ASPECT qsx qsx.aspect
9. MAP CALCULATOR
       qsx.dx =
       qsy.dy =
       erdep  = qsx.dx + qsy.dy
2.3 Notes

It is important to note that the algorithms available in ARC for interpolation and topographic analysis are less sophisticated that those in GRASS, therefore to obtain acceptable result more attention should be paid to the proper selection of resolution and to the quality of DEM.

Soil and cover parameters similar to those used in USLE or WEPP were not developed for the USPED model, because no systematic experimental work was performed. However, it was suggested (Moore and Wilson 1992) that under certain conditions, there is a relation between the USPED concept and USLE. It is therefore possible to combine the model with the USLE/RUSLE factors to predict the relative impact of land use change on erosion/deposition pattern assuming the conditions valid for USPED hold.  Caution should be used when interpreting the results because the USLE parameters were developed for simple plane fields and to obtain accurate predictions for complex terrain conditions they need to be re-calibrated ( Foster 1990, Mitasova et al 1997 reply).

3. References and links to related sites
 
Mitasova, H., Mitas, L., Brown, W. M., Johnston, D., 1998,
Multidimensional Soil Erosion/deposition Modeling and visualization
using GIS.
Final report for USA CERL. University of Illinois, Urbana-Champaign, IL.

Mitasova, H., J. Hofierka, M. Zlocha, L.R. Iverson, 1996,
       Modeling topographic potential for erosion and deposition using GIS.</a>
       Int. Journal of Geographical Information Science, 10(5), 629-641.
       (reply to a comment to this paper appears in 1997 in
       Int. Journal of Geographical Information Science, Vol. 11, No. 6)

Mitasova 1996 Mitasova, H., L. Mitas, B.M. Brown, D.P. Gerdes, I. Kosinovsky, 1995,
       Modeling spatially and temporally distributed phenomena: New methods
       and tools for GRASS GIS. </a>
       International Journal of GIS, 9 (4),
       special issue on integration of Environmental modeling and GIS, p. 443-446.

Mitasova et al. 1997,98,99 Applications presented at this WWW site:

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