Digital Elevation Model based
Hydrologic Modeling
Outline
• Topography and Physical runoff
generation processes (TOPMODEL)
• Raster calculation of wetness index
• Raster calculation of TOPMODEL runoff
• Extendability of ArcGIS using Visual
Basic Programming
Physical Processes involved in Runoff Generation
From http://snobear.colorado.edu/IntroHydro/geog_hydro.html
Runoff generation processes
Infiltration excess overland flow
aka Horton overland flow
P
qo
P
f
P
f
Partial area infiltration excess
overland flow
P
qo
P
P
f
Saturation excess overland flow
P
qo
P
qs
qr
P
Map of saturated areas showing expansion
during a single rainstorm. The solid black
shows the saturated area at the beginning of
the rain; the lightly shaded area is saturated
by the end of the storm and is the area over
which the water table had risen to the ground
surface. [from Dunne and Leopold, 1978]
Seasonal variation in pre-storm
saturated area [from Dunne and
Leopold, 1978]
Runoff generation at a point depends on
•
•
•
•
•
Rainfall intensity or amount
Antecedent conditions
Soils and vegetation
Depth to water table (topography)
Time scale of interest
These vary spatially which suggests a spatial
geographic approach to runoff estimation
TOPMODEL
Beven, K., R. Lamb, P. Quinn, R. Romanowicz and J. Freer, (1995),
"TOPMODEL," Chapter 18 in Computer Models of Watershed Hydrology, Edited
by V. P. Singh, Water Resources Publications, Highlands Ranch, Colorado, p.627668.
“TOPMODEL is not a hydrological modeling
package. It is rather a set of conceptual tools that
can be used to reproduce the hydrological
behaviour of catchments in a distributed or semidistributed way, in particular the dynamics of
surface or subsurface contributing areas.”
TOPMODEL and GIS
• Surface saturation and soil moisture deficits
based on topography
– Slope
– Specific Catchment Area
– Topographic Convergence
• Partial contributing area concept
• Saturation from below (Dunne) runoff
generation mechanism
Saturation in zones of convergent topography
Specific catchment area a is the
Numerical Evaluation
upslope area per unit contour
with the D Algorithm
2
length [m /m  m]
Steepest direction
Proportion
downslope
Topographic Definition
flowing to
neighboring
grid cell 4 is
1/(1+2)
Stream line
Proportion flowing to
neighboring grid cell 3
is 2/(1+2)
3
Contour line
4
2
1
2
Flow
direction.
5
1
6
8
7
Tarboton, D. G., (1997), "A New Method for the Determination of Flow Directions and
Contributing Areas in Grid Digital Elevation Models," Water Resources Research,
33(2): 309-319.) (http://www.engineering.usu.edu/cee/faculty/dtarb/dinf.pdf)
Hydrological processes within a
catchment are complex, involving:
•
•
•
•
Macropores
Heterogeneity
Fingering flow
Local pockets of saturation
The general tendency of water to flow
downhill is however subject to
macroscale conceptualization
TOPMODEL assumptions
• The dynamics of the saturated zone can be approximated
by successive steady state representations.
• The hydraulic gradient of the saturated zone can be
approximated by the local surface topographic slope, tan.
• The distribution of downslope transmissivity with depth is
an exponential function of storage deficit or depth to the
water table
T  To e S / m
-
T  To e  fz
To is lateral transmissivity [m2/h]
S is local storage deficit [m]
z is local water table depth [m] (=S/ne)
ne is effective porosity
m is a storage-discharge sensitivity parameter [m]
f =ne/m is an alternative storage-discharge sensitivity
parameter [m-1]
Topmodel - Assumptions
• The soil profile at each point
has a finite capacity to transport
water laterally downslope.
q cap  T S where T   K dz
e.g. T  KD
or
D
S
Dw

T   K oe
0
Units
D m
z m
K m/hr
f m-1
T
S
q
fz
Ko
dz 
f
m2/hr
dimensionless
m2/hr = m3/hr/m
Topmodel - Assumptions
Specific catchment area a [m2/m  m]
(per unit contour length)
• The actual lateral discharge is
proportional to specific
catchment area.
q act  R a
• R is
– Proportionality constant
– may be interpreted as “steady state”
recharge rate, or “steady state” per
unit area contribution to baseflow.
D
S
Dw
Units
a m
R m/hr
qact m2/hr = m3/hr/m
Topmodel - Assumptions
Specific catchment area a [m2/m  m]
(per unit coutour length)
• Relative wetness at a point and
depth to water table is
determined by comparing qact
and qcap
q act R a
w

q cap T S
D
S
• Saturation when w > 1.
a
1
i.e.

TS R
Dw
Topmodel
Specific catchment area a [m2/m  m]
(per unit coutour length)
a/TS or a/S or ln(a/S) or ln(a/tan)
[tan=S] is a wetness index that determines
the locations of saturation from below and
soil moisture deficit.
With uniform K and finite D assumption
Ra
a /S
1
w
w
where  '   a / S dA
TS
'
A
z  D(1  w )
z
D
S
Dw
With exponential K assumption
1  aR 
1 a
z   ln   z   ln    where
f  TS 
f S

1
1
R
   lna / S dA and z   (  ln )
A
f
T
Soil moisture deficit = z times porosity
Slope
Specific Catchment Area
Wetness Index ln(a/S)
from Raster Calculator.
Average,  = 6.91
Numerical Example
Given
• Ko=10 m/hr
• f=5 m-1
• Qb = 0.8 m3/s
• A (from GIS)
• ne = 0.2
Compute
• R=0.0002 m/h
• =6.90
• T=2 m2/hr
z  0.46 m
1 a
z  z   ln   
f S

Raster calculator -( [ln(sca/S)] - 6.90)/5+0.46
Depth to saturation z
Flat (0.5%)
-3 - 0 (7.8%)
0 - 0.1 (2.5%)
0.1 - 0.2 (4.0%)
0.2 - 0.5 (29%)
0.5 - 1 (56%)
1 - 1.5 (0.2%)
Calculating Runoff from 25 mm Rainstorm
• Flat area’s and z <= 0
– Area fraction (81 + 1246)/15893=8.3%
– All rainfall ( 25 mm) is runoff
• 0 < z  rainfall/effective porosity = 0.025/0.2 = 0.125 m
– Area fraction 546/15893 = 3.4%
– Runoff is P-z*0.2
– (1 / [Sat_during_rain ]) * (0.025 - (0.2 * [z]))
– Mean runoff 0.0113 m =11.3 mm
• z > 0.125 m
– Area fraction 14020/15893 = 88.2 %
– All rainfall infiltrates
• Area Average runoff
– 11.3 * 0.025 + 25 * 0.083 = 2.47 mm
– Volume = 0.00247 * 15893 * 30 * 30 = 35410 m3
Why Programming
GIS estimation of hydrologic
response function
• Amount of runoff generated
• Travel time to outlet
• Distance from each grid cell to outlet along
flow path (write program to do this)
• Distance from each point on contributing
area
– overlay grid to outlet distances with
contributing area.
Steps for distance to outlet program
• Read the outlet coordinates
• Read the DEM flow direction grid. This is a set of
integer values 1 to 8 indicating flow direction
• Initialize a distance to outlet grid with a no data
value
• Convert outlet to row and column references
• Start from the outlet point. Set the distance to 0.
• Examine each neighboring grid cell and if it drains
to the current cell set its distance to the outlet as
the distance from it to the current cell plus the
distance from the current cell to the outlet.
4
Programming
the calculation
of distance to
the outlet
3
5
6
2
1
7
8
Direction encoding
1
1
2
30
3
0
2
3
72.4 102.4
7
6
5
42.4 72.4
7
6
5
6
7
7
Distances to outlet
Recursive Procedure DISTANCE(i,j)
do for each neighbor (location in, jn)
If neighbor (in, jn) drains to cell (i,j)
Distance from (in, jn) is distance from (i,j) +
distance between cells (accounting for possible
diagonals)
Call recursive procedure on the neighbor,
DISTANCE(in, jn)
endif
end do
Visual Basic Programming in ArcMAP
References
ESRI, (1999), ArcObjects Developers Guide:
ArcInfo 8, ESRI Press, Redlands,
California.
Zeiler, M., (2001), Exploring ArcObjects. Vol
1. Applications and Cartography. Vol 2.
Geographic Data Management, ESRI,
Redlands, CA.
Are there any questions
?
AREA 2
3
AREA 1
12