While there are numerical landscape evolution models
that simulate how steady-state flows of water and sediment
reshape topography over long periods of time,
r.sim.terrain is the first to
simulate short-term topographic change
for both steady-state and dynamic flow regimes
across a range of spatial scales.
This free and open-source
Geographic Information Systems (GIS)-based topographic evolution model
uses empirical models for soil erosion
and a physics-based model
for shallow overland water flow and soil erosion
to compute short-term topographic change.
This model uses either a steady-state
or unsteady representation of overland flow
to simulate how overland sediment mass flows reshape topography
for a range of hydrologic soil erosion regimes
based on topographic, land cover, soil, and rainfall parameters.
As demonstrated by a case study
for the Patterson Branch subwatershed
on the Fort Bragg military installation in North Carolina,
r.sim.terrain simulates the development of
fine-scale morphological features including
ephemeral gullies, rills, and hillslopes.
Applications include land management, erosion control,
landscape planning, and landscape restoration.
Introduction
Landscape evolution models represent how the surface of the Earth changes
over time in response to physical processes.
Most studies of landscape evolution have been descriptive,
but a number of numerical landscape evolution models
have been developed that simulate elevational change over time
. Numerical landscape evolution models such as the
Geomorphic – Orogenic Landscape Evolution Model (GOLEM)
,
CASCADE ,
the Channel-Hillslope Integrated Landscape Development (CHILD) model
,
CAESAR ,
SIBERIA ,
LAPSUS
r.landscape.evol ,
and eSCAPE
simulate landscape evolution driven primarily by steady-state flows over long temporal scales.
(http://landlab.github.io/, last access: 3 July 2019),
a new Python library for numerically modeling Earth surface processes
,
has components for simulating landscape evolution such as the
Stream Power with Alluvium Conservation and Entrainment (SPACE)
model .
While Geographic Information Systems (GIS)
support efficient data management,
spatial and statistical modeling and analysis,
and visualization,
there are few GIS-based soil erosion models (see Table )
or landscape evolution models.
developed the model r.terradyn
as a Geographic Resources Analysis Support System (GRASS) GIS shell script module
to simulate terrain evolution
by steady-state net erosion–deposition rates
estimated by the Simulation of Water Erosion (SIMWE) model
and gravitational diffusion.
developed a long-term landscape evolution model
in GRASS GIS called r.landscape.evol that integrates
the Unit Stream Power Erosion Deposition (USPED) model,
fluvial erosion, and gravitational diffusion.
r.landscape.evol has been used to simulate the impact
of prehistoric settlements on Mediterranean landscapes.
In spite of the recent progress in landscape evolution modeling and monitoring,
there are still major research questions
to address in the theoretical foundations of erosion modeling
such as how erosional processes scale over time and space,
and how sediment detachment and transport interact .
While most numerical landscape evolution models
simulate erosion processes at steady-state peak flows,
short-term erosional processes like gully formation
can be driven by unsteady, dynamic flow
with significant morphological changes happening before flows reach steady state.
A landscape evolution model with dynamic water and sediment flow
is needed to study fine-scale spatial
and short-term erosional processes
such as gully formation and the development of microtopography.
Examples of geospatial soil erosion models.
ModelSpatial scaleTemporal scaleRepresentationImplementationReferenceRUSLE3Dregionalcontinuousrastermap algebraUSPEDwatershedcontinuousrastermap algebraSIMWEwatershedevent – continuousrasterGRASS GIS modules https://grass.osgeo.org/grass74(last access: 3 July 2019)GeoWEPPwatershedcontinuousrasterArcGIS module http://geowepp.geog.buffalo.edu/(last access: 3 July 2019)AGWAwatershedevent – continuousvectorArcGIS module https://www.tucson.ars.ag.gov/agwa/(last access: 3 July 2019)openLISEMwatershedeventrasterPCRaster script https://blog.utwente.nl/lisem/(last access: 3 July 2019)Landlabwatershedevent – continuousraster + meshPython library https://github.com/landlab/(last access: 3 July 2019)
At the beginning of a rainfall event,
overland water flow is unsteady –
its depth changes at a variable rate over time and space.
If the intensity of rainfall continues to change throughout the event,
then the flow regime will remain dynamic.
If, however, overland flow reaches a peak rate,
then the hydrologic regime is considered to be at steady state.
At steady state,
∂h(x,y,t)∂t=0,
where (x,y) is the position [m], t is the time [s], and
h(x,y,t) is the depth of overland flow [m].
Gullies are eroded, steep-banked channels formed by ephemeral, concentrated flows of water. A gully forms when overland water flow
converges in a knickzone – a concave space with steeper slopes than its surroundings
–
during intense rainfall events.
When the force of the water flow concentrated in the knickzone
is enough to detach and transport large amounts of sediment,
an incision begins to form at the apex of the knickzone
– the knickpoint or headwall.
As erosion continues, the knickpoint begins to migrate upslope
and the nascent gully channel widens,
forming steep channel banks.
Multiple incisions initiated by different knickpoints
may merge into a gully channel
and multiple channels may merge
into a branching gully system .
This erosive process is dynamic;
the morphological changes drive further changes
in a positive feedback loop.
When the gully initially forms,
the soil erosion regime should be detachment capacity limited
with the concentrated flow of water in the channel of the gully
detaching large amounts of sediment
and transporting it to the foot of the gully,
potentially forming a depositional fan.
If the intensity of rainfall decreases
and transport and detachment capacity
approach a balance,
then the soil erosion regime may switch to
a variable erosion–deposition regime,
in which soil is eroded and deposited
in a spatially variable pattern.
Subsequent rainfall events may trigger further
knickpoint formation and upslope migration,
channel incision and widening, and
depositional fan and ridge formation.
Between high-intensity rainfall events,
lower-intensity events and gravitational diffusion
may gradually smooth the shape of the gully.
Eventually, if detachment capacity
significantly exceeds transport capacity
and the regime switches to transport capacity limited,
the gully may fill with sediment
as soil continues to be eroded
but cannot be transported far.
Gully erosion rates and evolution
can be monitored in the field
or modeled on the computer.
Field methods include
dendrogeomorphology and
permanent monitoring stakes for recording erosion rates,
extensometers for recording mass wasting events,
weirs for recording water and suspended sediment discharge rates,
and time series of surveys using
total station theodolites ,
unmanned aerial systems (UASs) ,
airborne lidar ,
and terrestrial lidar .
With terrestrial lidar, airborne lidar, and
UAS photogrammetry,
there are now sufficient-resolution topographic data
to morphometrically analyze and
numerically model fine-scale landscape evolution in GIS,
including processes such as gully formation
and the development of microtopography.
Gully erosion has been simulated with
RUSLE2-Raster (RUSLER)
in conjunction with the Ephemeral Gully Erosion Estimator (EphGEE)
,
while gully evolution
has been simulated for detachment-capacity-limited erosion regimes
with the Simulation of Water Erosion (SIMWE) model
.
Now numerical landscape evolution models
that can simulate
steady-state and unsteady flow regimes
and can dynamically switch between soil erosion regimes
are needed to study
fine-scale spatial and short-term erosional processes.
The numerical landscape evolution model
r.sim.terrain was developed to
simulate the spatiotemporal evolution of landforms
caused by shallow overland water and sediment flows
at spatial scales ranging from
square meters to kilometers
and temporal scales ranging from minutes to years.
This open-source GIS-based landscape evolution model can
simulate either steady-state or unsteady flow regimes,
dynamically switch between soil erosion regimes, and
simulate the evolution of fine-scale morphological features
such as ephemeral gullies
(Fig. ).
It was designed as a research tool for
studying how erosional processes scale over time and space,
comparing empirical and process-based models,
comparing steady-state and unsteady flow regimes, and
studying the role of unsteady flow regimes
in fine-scale morphological change.
r.sim.terrain was tested with
a subwatershed scale (450 m2) case study
and the simulations were compared against
a time series of airborne lidar surveys.
The digital elevation model (DEM) (a) before
and (b) after simulated landscape evolution with r.sim.terrain for a
subwatershed of Patterson Branch, Fort Bragg, NC, USA. The before DEM was generated
from an airborne lidar data acquired in 2012. The simulation used the SIMWE model for
a 120 min rainfall event with 50 mm h-1 for a variable erosion–deposition
regime at steady state. In the evolved DEM, the gully channel has widened with
depositional ridges forming along its thalweg.
r.sim.terrain
The process-based, spatially distributed landscape evolution model r.sim.terrain simulates topographic changes caused by shallow, overland water flow across a range of spatiotemporal scales and soil erosion regimes using either the Simulated Water Erosion (SIMWE) model, the 3-Dimensional Revised Universal Soil Loss Equation (RUSLE3D) model, or the USPED
model (Fig. ).
The r.sim.terrain model
can simulate either steady-state or dynamic flow regimes.
SIMWE is a physics-based simulation
that uses a Monte Carlo path sampling method
to solve the water and sediment flow equations
for detachment-limited, transport-limited, and variable erosion–deposition
soil erosion regimes
.
With SIMWE,
r.sim.terrain
uses the modeled flow of sediment
– a function of water flow and soil detachment and transport parameters –
to estimate net erosion and deposition rates.
RUSLE3D is an empirical equation for estimating soil erosion rates
in detachment-capacity-limited soil erosion regimes
.
With RUSLE3D, r.sim.terrain
uses an event-based rainfall erosivity factor,
soil erodibility factor, land cover factor, and 3-D topographic factor
– a function of slope and flow accumulation –
to model soil erosion rates.
USPED is a semi-empirical equation for net erosion and deposition
in transport-capacity-limited soil erosion regimes
.
With USPED, r.sim.terrain uses an event-based rainfall erosivity factor,
soil erodibility factor, land cover factor, and a topographic sediment transport factor
to model net erosion or deposition rates as the divergence of sediment flow.
For each of the models,
topographic change is derived at each time step
from the net erosion–deposition rate
and gravitational diffusion.
Depending on the input parameters,
r.sim.terrain simulations with SIMWE
can represent variable soil erosion–deposition regimes,
including prevailing detachment-capacity-limited
or prevailing transport-capacity-limited regimes.
Conceptual
diagram for r.sim.terrain.
The r.sim.terrain model
can simulate the evolution of gullies
including processes such as
knickpoint migration,
channel incision,
channel widening,
aggradation,
scour pit formation,
depositional ridge formation
along the thalweg of the gully,
and depositional fan formation at the foot of the gully.
Applications include
geomorphological research,
erosion control,
landscape restoration,
and scenario development
for landscape planning and management.
This model can simulate landscape evolution
over a wide range of spatial scales
from small watersheds
less than 10 km2 with SIMWE
to regional watersheds
of 100 km2
with USPED or RULSE3D,
although it does not model fluvial processes.
It has been used at resolutions ranging from submeter scale to 30 m.
The model has been implemented
as a Python add-on module
for the free, open-source GRASS GIS (https://grass.osgeo.org/, last access: 3 July 2019)
.
The source code is available at https://github.com/baharmon/landscape_evolution (last access: 3 July 2019)
under the GNU General Public License v2 .
It supports multithreading and parallel processing
to efficiently compute simulations
using large, high-resolution topographic datasets.
The landscape evolution model
can be installed in GRASS GIS as an add-on module
with the command :
Landscape evolution
Landscape evolution in r.sim.terrain
is driven by change in the elevation surface
caused by soil erosion and deposition.
During storm events, overland flow erodes soil and
transports sediment across landscape, and
under favorable conditions deposits the sediment.
Gravitational diffusion,
applied to the changed elevation surface,
simulates the smoothing effects
of localized soil transport between events.
Elevation change
Assuming negligible uplift, the change in elevation over time
is described by the continuity of mass equation
expressed as the divergence of sediment flow :
∂z∂t=-∇⋅qsρs-1=dsρs-1,
where z is elevation [m], t is time [s], qs
is sediment flow per unit width (vector) [kg m-1 s-1], ds is the net erosion–deposition rate [kg m-2 s-1], and ρs is sediment mass density [kg m-3].
In r.sim.terrain,
the net erosion–deposition rate ds driven by overland flow
is estimated at different levels of complexity based
on the simulation mode selected by the user.
Gravitational diffusion is then applied to the changed topography
to simulate the smoothing effects
of localized soil transport between rainfall events.
The change in elevation due to gravitational diffusion
is a function of the diffusion coefficient and the Laplacian of elevation
:
∂z∂t=εg∇2z,
where εg is the diffusion coefficient [m2 s-1].
The discrete implementation follows :
4zt+Δt1=zt+Δzs5zt+Δt1+Δt2=zt+Δt1+Δzg,
where Δzs is elevation change [m] caused by net erosion or deposition during time interval Δt1
(Eq. ), and Δzg is the diffusion-driven elevation change [m] during time interval Δt1
(Eq. ).
Erosion–deposition regimes
Following experimental observations and qualitative arguments,
proposed that the sum of
the ratio of the net erosion–deposition rate ds
to the detachment capacity
Dc [kg m-2 s-1]
and the ratio of the sediment flow rate qs=|qs| to
the sediment transport capacity Tc [kg m-1 s-1]
is a conserved quantity (unity):
dsDc+qsTc=1.
The net erosion and deposition rate ds can then be expressed
as being proportional to the difference between
the sediment transport capacity Tc
and the actual sediment flow rate qs:
ds=DcTc(Tc-qs).
This principle is used in several erosion models
including the Water Erosion Prediction Project (WEPP)
and SIMWE .
Using this concept, it is possible to identify
two limiting erosion–deposition regimes.
When Tc≫Dc leading to Tc≫qs,
the erosion regime is detachment capacity limited and
net erosion is equal to the detachment capacity:
ds=Dc.
For this case, the transport capacity of overland flow
exceeds the detachment capacity,
and thus sediment flow, erosion, and sediment transport
are limited by the detachment capacity.
Therefore, no deposition occurs.
An example of this case is when a strong storm
producing intense
overland flow over compacted clay soils
causes high-capacity flows to transport light clay particles,
while the detachment of compacted soils is limited.
When Dc≫Tc, sediment flow is at sediment transport capacity qs=Tc,
leading to a transport-capacity-limited regime
with deposition reaching its maximum extent for the given water flow.
Net erosion–deposition is computed as the divergence of
transport capacity multiplied by a unit vector s0
in the direction of flow:
ds=∇⋅Tcs0.
This case may occur, for example, during a moderate storm
with overland flow over sandy soils
with high detachment capacity but low transport capacity.
For 0<(Dc/Tc)<∞,
the spatial pattern of net erosion–deposition is variable
and depends on the difference between the sediment transport capacity
and the actual sediment flow rate at the given location.
The detachment capacity Dc and the sediment transport capacity Tc
are estimated using shear stress and stream power equations, respectively,
expressed as power functions of water flow properties and slope angle.
The relations between the topographic parameters
of well-known empirical equations for erosion modeling,
such as the Universal Soil Loss Equation (USLE) and stream power, were presented by
and used to develop simple, GIS-based models for limiting erosion–deposition cases
such as RUSLE3D and USPED .
The SIMWE model estimates Tc and Dc using modified
equations and parameters developed for the WEPP model
.
The simulation modes in r.sim.terrain include (Fig. )
the process-based SIMWE model for steady-state and unsteady shallow
overland flow in variable erosion–deposition regimes with ds
computed by solving the shallow water flow and sediment transport continuity
equations,
the RUSLE3D model for detachment-capacity-limited cases with ds given by Eq. (), and
the USPED model for transport-capacity-limited regimes
with ds given by Eq. ().
The following sections explain the computation of ds for these three modes in more detail.
Simulation of Water Erosion (SIMWE)
SIMWE is a physics-based simulation of shallow overland water and sediment flow
that uses a path sampling method to solve the continuity equations
with a 2-D diffusive wave approximation
.
SIMWE has been implemented in GRASS GIS as the modules
r.sim.water
and r.sim.sediment.
In SIMWE mode, for each landscape evolution time step,
r.sim.terrain
computes the first-order partial derivatives of the elevation surface
∂z/∂x and ∂z/∂y,
simulates shallow water flow depth, sediment flow, and the net erosion–deposition rate, and
then evolves the topography based on the erosion–deposition rate and gravitational diffusion.
The first-order partial derivatives of the elevation surface
are computed using the GRASS GIS module r.slope.aspect
using the equations in .
r.sim.terrain simulates unsteady-state flow regimes
when the landscape evolution time step is less than the travel time
for a drop of water or a particle of sediment to cross the landscape,
e.g., when the time step is less than the
time to concentration for the modeled watershed.
With longer landscape evolution time steps,
the model simulates a steady-state regime.
Shallow water flow
The SIMWE model simulates shallow overland water flow
controlled by spatially variable topographic, soil, land cover,
and rainfall parameters using a Green function Monte Carlo path sampling method.
The steady-state shallow water flow continuity equation relates the change in water depth across space
to source, defined in our case as rainfall excess rate:
∇⋅q=∇⋅(hv)=∇⋅n-1h5/3s1/2s0=ie,
where q is the water flow per unit width (vector) [m2 s-1], h is the depth of overland flow [m], v is the water flow velocity vector [m s-1] whose magnitude is computed with Manning's equation v=n-1h2/3s1/2, n is Manning's coefficient [s m-1/3], s is slope (unitless), and ie is the rainfall excess rate [m s-1]
(i.e., rainfall intensity - infiltration - vegetation intercept).
An approximation of diffusive wave effects is incorporated by adding a diffusion term proportional to
∇2[h5/3]:
-εw2∇2h5/3+∇⋅n-1h5/3s1/2s0=ie,
where εw is a spatially variable diffusion coefficient [m4/3 s-1].
The path sampling method solves the continuity equation for h5/3
through the accumulation of the evolving source .
The solution assumes that water flow velocity
is largely controlled by the slope of the terrain and surface roughness
and that its change at a given location during the simulated event is negligible.
The initial number of particles per grid cell is proportional to the rainfall excess rate ie (source).
The water depth h5/3 at time τ during the simulated rainfall event
is computed as a function of particle (walkers) density at each grid cell.
Particles are routed across the landscape
by finding a new position for each walker at time τ+Δτ:
rmnew=rm+Δτv+g,
where r=(x,y) is the mth walker position [m], Δτ is the particle routing time step [s], and g is a random vector with Gaussian components with variance Δτ [m].
The mathematical background of the method,
including the computation
of the temporal evolution of water depth
and incorporation of approximate momentum
through an increased diffusion rate in the prevailing direction of flow,
is presented by and .
Sediment flow and net erosion–deposition
The SIMWE model simulates the sediment flow over complex topography
with spatially variable overland flow, soil, and land cover properties
by solving the sediment flow continuity equation
using a Green function Monte Carlo path sampling method.
Steady-state sediment flow qs is approximated by
the bivariate continuity equation, which relates
the change in sediment flow rate to effective sources and sinks:
∇⋅qs=sources-sinks=ds.
The sediment flow rate qs
is a function of water flow and sediment concentration
:
qs=ρscq=ρschv=ϱv,
where ρs is sediment mass density in the water column [kg m-3], c is sediment concentration [particle m-3], and ϱ=ρsch is the mass of sediment transported by water per unit area [kg m-2].
The sediment flow equation (Eq. ),
like the water flow equation,
has been rewritten to include a small diffusion term that is
proportional to the mass of water-carried sediment per unit area
∇2ϱ:
-εs2∇2ϱ+∇⋅(ϱv)+ϱDcTc|v|=ds,
where εs is the diffusion constant [m2 s-1].
On the left-hand side of Eq. (),
the first term describes local diffusion,
the second term is drift driven by water flow,
and the third term represents a velocity-dependent “potential”
acting on the mass of transported sediment.
The initial number of particles per grid cell
is proportional to the soil detachment
capacity Dc (source).
The particles are then routed across the landscape
by finding a new position for each walker at time τ+Δτ:
rmnew=rm+Δτv+g,
while the updated weight is
wmnew=wmexp-Δτurmnew+urm/2,
where u=Dc/Tc|v|.
Sediment flow is computed
as the product of weighted particle densities
and the water flow velocity (Eq. ), and
the net erosion–deposition rate ds
is computed as the divergence of sediment flow using Eq. ().
See and for more details on the Green function
Monte Carlo solution and equations for computing Dc and Tc.
This model can simulate erosion regimes from prevailing detachment-limited conditions when Tc≫Dc
to prevailing transport-capacity-limited conditions when Dc≫Tc
and the erosion–deposition patterns between these conditions.
At each landscape evolution time step, the regime can change based on
the ratio between the sediment detachment capacity Dc
and the sediment transport capacity Tc and the actual sediment flow rate.
If the landscape evolution time step is shorter than the time to concentration
(i.e., the time for water to reach steady state),
then net erosion–deposition is derived from unsteady flow.
Revised Universal Soil Loss Equation for Complex Terrain (RUSLE3D)
RUSLE3D
is an empirical model for computing erosion
in a detachment-capacity-limited soil erosion regime
for watersheds with complex topography .
It is based on
the USLE,
an empirical equation for estimating the average
sheet and rill soil erosion from rainfall and runoff
on agricultural fields and rangelands with simple topography
.
It models erosion-dominated regimes without deposition
in which sediment transport capacity is
uniformly greater than detachment capacity.
In USLE, soil loss per unit area is determined by
an erosivity factor R,
a soil erodibility factor K,
a slope length factor L,
a slope steepness factor S,
a cover management factor C,
and a prevention measures factor P.
These factors are empirical constants derived
from an extensive collection of measurements
on 22.13 m standard plots with an average slope of 9 %.
RUSLE3D was designed to account for more complex, 3-D topography
with converging and diverging flows.
In RUSLE3D, the topographic potential for erosion at any given point
is represented by a 3-D topographic factor LS3-D,
which is a function of the upslope contributing area
and the angle of the slope.
In this spatially and temporally distributed model,
RUSLE3D is modified by the use of a
event-based R factor derived from rainfall intensity at each time step.
For each time step, this model computes the parameters for RUSLE3D –
an event-based erosivity factor,
the slope of the topography, the flow accumulation, and
the 3-D topographic factor –
and then solves the RUSLE3D equation for the rate of soil loss
(i.e., the net soil erosion rate).
The soil erosion rate is then used to simulate landscape evolution
in a detachment-capacity-limited soil erosion regime.
Erosivity factor
The erosivity factor R in USLE and RUSLE
is the combination of the total energy
and peak intensity of a rainfall event,
representing the interaction
between the detachment of sediment particles
and the transport capacity of the flow.
It can be calculated as the product of the
the kinetic energy of the rainfall event E
and its maximum 30 min intensity I30.
In this model, however, the erosivity factor
is derived at each time step as a function of
kinetic energy, rainfall depth, rainfall intensity, and time.
First, rain energy is derived from rainfall intensity
:
ere0=1.-bexpiri0,
where er is unit rain energy [MJ ha-1 mm-1],
ir is rainfall intensity [mm h-1], b is empirical
coefficient, i0 is reference rainfall intensity [mm h-1], and e0 is reference energy [MJ ha-1 mm-1]. The parameters for this equation were derived from observed data
published for different regions by . Then the event-based erosivity index Re
is calculated as the product of
unit rain energy, rainfall depth, rainfall intensity, and time:
Re=ervrirΔt,
where Re is the event-based erosivity index [MJ mm ha-1 h-1], vr is the rainfall depth [mm] derived from vr=irΔt, and Δt is the change in time [s].
Flow accumulation
The upslope contributing area per unit width a
is determined by flow accumulation
(the number of grid cells draining into a given grid cell)
multiplied by grid cell width (Fig. d).
Flow accumulation is calculated using
a multiple flow direction algorithm
based on AT least-cost path searches .
The multiple flow direction algorithm
implemented in GRASS GIS, as the module r.watershed
is computationally efficient, does not require sink filling
and can navigate nested depressions and other obstacles.
Water and sediment flows modeled with spatially variable land cover for
Patterson Branch, Fort Bragg, NC: (a) water depth [m] simulated by SIMWE
for a 10 min event with 50 mm h-1 in the subwatershed; (b) flow
accumulation for RUSLE3D in the subwatershed; (c) erosion and
deposition [kg m-2 s-1] simulated by SIMWE in drainage area 1; and
(d) erosion [kg m-2 s-1] modeled by RUSLE3D in drainage area 1.
3-D topographic factor
The 3-D topographic factor LS3-D
is calculated as a function of the upslope contributing area
and the slope (Fig. e):
LS3-D=(m+1)aa0msinββ0n,
where LS3-D is the dimensionless topographic factor, a is upslope contributing area per unit width [m], a0 is the length of the standard USLE plot [22.1 m], β is the angle of the slope [∘], m is an empirical coefficient, n is an empirical coefficient, and β0 is the slope of the standard USLE plot [5.14∘].
The empirical coefficients m and n
for the upslope contributing area and the slope
can range from 0.2 to 0.6 and 1.0 to 1.3, respectively,
with low values representing dominant sheet flow
and high values representing dominant rill flow.
Detachment-limited erosion rate
The erosion rate is a function of the event-based erosivity factor,
soil erodibility factor, 3-D topographic factor,
land cover factor, and prevention measures factor
(Fig. d):
E=ReKLS3-DCP,
where E is soil erosion rate (soil loss) [kg m-2 min-1], Re is the event-based erosivity factor [MJ mm ha-1 h-1], K is the soil erodibility factor [t ha h ha-1 MJ-1 mm-1], LS3-D is the dimensionless topographic (length–slope) factor, C is the dimensionless land cover factor, and P is the dimensionless prevention measures factor.
The detachment-limited erosion represented by RUSLE3D leads to the simulated change in elevation:
Δzs=Dcρs-1=Eρs-1,
which is combined with Eq. () for gravitational diffusion.
Unit Stream Power Erosion Deposition (USPED)
USPED estimates net erosion–deposition
as the divergence of sediment flow
in a transport-capacity-limited soil erosion regime.
The amount of soil detached is
close to the amount of sediment that water flow can carry.
As a transport-capacity-limited model,
USPED predicts erosion where transport capacity increases
and deposition where transport capacity decreases.
The influence of topography on sediment flow
is represented by a topographic sediment transport factor,
while the influence of soil and land cover is represented by
factors adopted from USLE and RUSLE
.
Sediment flow is estimated by computing
the event-based erosivity factor (Re)
using Eq. (),
the slope and aspect of the topography,
the flow accumulation with a multiple flow direction algorithm,
the topographic sediment transport factor,
and sediment flow at transport capacity.
Net erosion–deposition is then computed as the divergence of sediment flow.
Topographic sediment transport factor
Using the unit stream power concept presented by ,
the 3-D topographic factor (Eq. )
for RUSLE3D is modified to represent
the topographic sediment transport factor (LST) –
the topographic component
of overland flow at sediment transport capacity:
LST=amsinβn,
where LST is the topographic sediment transport factor, a is the upslope contributing area per unit width [m], β is the angle of the slope [∘], m is an empirical coefficient, and n is an empirical coefficient.
Transport-limited sediment flow and net erosion–deposition
Sediment flow at transport capacity is a function of
the event-based rainfall factor, soil erodibility factor,
topographic sediment transport factor,
land cover factor, and prevention measures factor:
T=ReKCPLST,
where T is sediment flow at transport capacity [kg m-1 s-1], Re is the event-based rainfall factor [MJ mm ha-1 h-1], K is the soil erodibility factor [t ha h ha-1 MJ-1 mm-1], C is the dimensionless land cover factor, and P is the dimensionless prevention measures factor.
Net erosion–deposition is estimated as the divergence of sediment flow,
assuming that sediment flow is equal to sediment transport capacity:
ds=∂Tccosα∂x+∂Tcsinα∂y,
where ds is net erosion–deposition [kg m-2 s-1], α is the aspect of the topography (i.e., the direction of flow) [∘]. With USPED, the simulated change in elevation Δzs=ds
is derived from Eq. () for landscape evolution
and then Eq. ()
for gravitational diffusion.
Case study
Military activity is a high-impact land use
that can cause significant physical alteration to the landscape.
Erosion is a major concern for military installations,
particularly at training bases,
where the land surface is disturbed by
off-road vehicles, foot traffic, and munitions.
Off-road vehicles and foot traffic by soldiers
cause the loss of vegetative cover,
the disruption of soil structure, soil compaction,
and increased runoff due to
reduced soil capacity for water infiltration
.
Gullies – ephemeral channels with steep headwalls
that incise into unconsolidated soil to depths of meters –
are a manifestation of erosion common to
military training installations like Fort Bragg in North Carolina
and the Piñon Canyon Maneuver Site in Colorado.
While the local development of gullies can restrict
the maneuverability of troops and vehicles during training exercises,
pervasive gullying across a landscape
can degrade an entire training area
.
To test the effectiveness of the different models
in r.sim.terrain,
we compared the simulated evolution
of a highly eroded subwatershed
of Patterson Branch on Fort Bragg, North Carolina,
against a time series of airborne lidar surveys.
The models – SIMWE, RUSLE3D, and USPED –
were tested in steady-state and dynamic modes
for design storms with constant rainfall.
Patterson Branch
With 650 km2 of land,
Fort Bragg is the largest military installation in the US
and has extensive areas of bare, erodible soils
on impact areas, firing ranges, landing zones, and drop zones.
It is located in the Sandhills region of North Carolina
with a longleaf pine and wiregrass ecosystem .
The study landscape
– a subwatershed of Patterson Branch (Fig. )
in the Coleman Impact Area –
is pitted with impact craters from artillery and mortar shells
and has an active, approximately 2 m deep gully.
It is a pine-scrub oak Sandhills community
composed primarily of longleaf pine (Pinus palustris)
and wiregrass (Aristida stricta)
on Blaney and Gilead loamy sands
.
Throughout the Coleman Impact Area,
frequent fires ignited by live munitions
drive the ecological disturbance regime
of this fire-adapted ecosystem.
In 2016, the 450 m2 study site was
43.24 % bare ground with predominately loamy sands,
39.54 % covered by the wiregrass community, and
17.22 % forested with the longleaf pine community
(Fig. a).
We hypothesize that the elimination of forest cover
in the impact zone
triggered extensive channelized overland flow,
gully formation, and sediment transport into the creek.
Subwatershed with 2014 orthoimagery
(a) draped over the 2016 digital elevation model
and (b) drainage areas with 2014 orthoimagery, Patterson Branch, Fort Bragg, NC, USA.
Morphological change in the subwatershed of Patterson Branch, Fort Bragg, NC, USA:
(a) land cover in 2014,
(b) landforms in 2012,
(c) elevation difference between 2012 and 2016 [m], and
(d) landforms in 2016.
Time series of digital elevation models
and land cover maps for the study landscape
were generated from lidar point clouds and orthophotography.
The digital elevation models for 2004, 2012, and 2016
were interpolated at 1 m resolution
using the regularized spline with tension function
from airborne lidar surveys
collected by the NC Floodplain Mapping Program and Fort Bragg.
Unsupervised image classification
was used to identify clusters of spectral reflectance
in a time series of 1 m resolution orthoimagery
collected by the National Agriculture Imagery Program.
The land cover maps were derived from the
classified lidar point clouds and the classified orthoimagery.
Spatially variable soil erosion factors
– the k factor, c factor, Manning's coefficient, and runoff rate –
were then derived from the land cover and soil maps.
The dataset for this study is hosted at
https://github.com/baharmon/landscape_evolution_dataset (last access: 3 July 2019)
under the ODC Open Database License (ODbL) .
The data are derived from publicly available data from
the US Army, USGS, USDA, Wake County GIS, NC Floodplain
Mapping Program, and the NC State Climate Office.
There are detailed instructions for preparing the input data in the
tutorial (https://github.com/baharmon/landscape_evolution/blob/master/tutorial.md, last access: 3 July 2019, )
and a complete record of the commands used to process the sample data in the
data log (https://github.com/baharmon/landscape_evolution_dataset/blob/master/nc_spm_evolution/DATA.md, last access: 3 July 2019, ).
We used the geomorphons method
of automated landform classification
based on the openness of terrain
and the difference between the digital elevation models
to analyze the changing morphology of the study area
(Figs. and ).
The 2 m deep gully –
its channels classified as valleys and
its scour pits as depressions by geomorphons –
has multiple mature branches
and ends with a depositional fan.
The gully has also developed
depositional ridges beside the channels.
Deep scour pits have developed
where branches join the main channel
and where the main channel has sharp bends.
A new branch has begun to form
in a knickzone classified as a mix of valleys and hollows
on a grassy swale on the northeast side of the gully.
Between 2012 and 2016 a depositional ridge
developed at the foot of this nascent branch
where it would meet the main channel.
The 2016 minus 2012 DEM of difference (DoD) –
i.e., the difference in elevation
(Figs. c and c) –
shows a deepening of the main channel
by approximately 0.2 m
and scours pits by approximately 1 m,
while depositional ridges have formed and grown up to
approximately 1 m high.
The DoD also shows that
244.60 m3 of sediment were deposited
on the depositional fan between 2012 and 2016.
Detailed morphological change
for drainage area 1 of Patterson Branch, Fort Bragg, NC, USA:
(a) land cover in 2014,
(b) landforms in 2012,
(c) elevation difference between 2012 and 2016 [m], and
(d) landforms in 2016.
Simulations
We ran a sequence of r.sim.terrain simulations
with design storms
for the Patterson Branch subwatershed study area
to demonstrate the capabilities
of the RUSLE3D, USPED, and SIMWE models
(Table ).
To analyze the results of the simulations,
we compared
net differences in elevation
morphological features,
and volumetric change.
While r.sim.terrain can use rainfall records,
we used design storms to demonstrate and test
the basic capabilities of the model.
Our design storms were based off the peak rainfall values
in records from the State Climate Office of North Carolina.
We used RUSLE3D to simulate landscape evolution
in a dynamic, detachment-capacity-limited soil erosion regime
for a 120 min design storm
with 3 min intervals
and a constant rainfall intensity of 50 mm h-1
(Fig. ).
We used USPED to simulate landscape evolution
in a dynamic, transport-capacity-limited soil erosion regime
for a 120 min design storm
with 3 min intervals
and a constant rainfall intensity of 50 mm h-1
(Fig. ).
We used SIMWE to simulate landscape evolution
in a steady state, variable erosion–deposition soil erosion regime
for a 120 min design storm
with a constant rainfall intensity of 50 mm h-1
(Fig. ).
In all of the simulations,
a sink-filling algorithm
– an optional parameter in r.sim.terrain –
was used to reduce the effects of positive feedback loops
that cause the overdevelopment of scour pits.
Landscape evolution simulations.
Flow regimeModelIntensityDurationIntervalmnρsThreadsRuntimeDynamicRUSLE3D50 mm h-1120 min3 min0.41.32 min 36 sDynamicUSPED50 mm h-1120 min3 min1.51.21.63 min 14 sSteady stateSIMWE50 mm h-1120 min120 min1.6644 min 51 s
Dynamic simulation with RUSLE3D for a 120 min event
with a rainfall intensity of 50 mm h-1 for Patterson Branch, Fort Bragg, NC:
(a) flow accumulation and
(b) erosion [kg m-2 s-1]
for the subwatershed in the final 3 min time step;
(c) net difference [m] and (d) landforms
for drainage area 1.
Dynamic simulation with USPED
for a 120 min event
with a rainfall intensity of 50 mm h-1
for Patterson Branch, Fort Bragg, NC:
(a) flow accumulation and
(b) erosion–deposition [kg m-2 s-1]
for the subwatershed in the final 3 min time step;
(c) net difference [m] and (d) landforms
for drainage area 1.
Steady-state SIMWE simulations
for a 120 min event
with a rainfall intensity of 50 mm h-1
for Patterson Branch, Fort Bragg, NC:
(a) depth of unsteady flow [m] and
(b) erosion–deposition [kg m-2 s-1]
for the subwatershed;
(c) net difference [m] and (d) landforms
for drainage area 1.
The simulations were automated and run in parallel
using Python scripts that are available in the
software repository (https://github.com/baharmon/landscape_evolution, last access: 3 July 2019, ).
The simulations can be reproduced using these scripts
and the study area dataset
by following the instructions
in the Open Science Framework repository
at https://osf.io/tf6yb/ (last access: 3 July 2019).
The simulations were run
in GRASS GIS 7.4
on a desktop computer
with 64 bit Ubuntu 16.04.4 LTS,
8×4.20 GHz Intel Core i7 7700K CPUs,
and 32 GB RAM.
Simulations using SIMWE
are far more computationally intensive
than RULSE3D or USPED
but support multithreading
when compiled with OpenMP.
Dynamic simulations of RUSLE3D and USPED took
2 min 36 s and
3 min 14 s, respectively,
to run on a single thread,
while the steady-state simulation for SIMWE took
44 min 51 s to run on six threads
(Table ).
Results
We used the difference in DEMs to compute volumetric changes
between the lidar surveys and the simulations
(Table ). We applied a threshold of ±0.18 m to the lidar surveys since they had a vertical accuracy at a 95 % confidence level of 18.15 cm
based on a 9.25 cm root mean square error in z (RMSEz)
for non-vegetated areas in accordance with
the National Digital Elevation Program guidelines .
Given the presence of the mature gully
with ridges along its banks,
we hypothesize that
the study landscape had previously been dominated by
a detachment-limited soil erosion regime
but – given the net change of 654.77 m3 –
had switched to a transport-capacity-limited or
variable erosion–deposition regime
during our study period.
Volumetric change.
Difference of DEMs (DoD)Threshold [m]Erosion [m3]Deposition [m3]Net change [m3]2016–2012±0.18152.96807.74654.77Simulated with RUSLE3D – 2012None1480.750-1480.75Simulated with USPED – 2012None1235.08727.46-507.62Simulated with SIMWE – 2012None758.56608.91-149.664
The dynamic RUSLE3D simulation carved a deep incision
in the main gully channel where water accumulated
(Fig. ). As a detachment-capacity-limited model,
RUSLE3D's results were dominated by erosion and
thus negative elevation change. It eroded 1480.75 m3 of sediment
with no deposition.
The dynamic USPED simulation
eroded the banks of the gully
and deposited in channels
causing the gully grow wider and shallower
(Fig. ).
As a transport-capacity-limited model,
USPED generated a distributed pattern
with both erosion and deposition.
Erosion far exceeded deposition with
1235.08 m3 of sediment eroded
and 727.46 m3 deposited
for a net change of -507.62 m3.
While USPED's pattern of elevation change
was grainy and fragmented,
it captured the process of channel
filling and widening expected with
a transport-capacity-limited soil erosion regime.
The steady-state SIMWE simulation
for a variable erosion–deposition regime
predicted the morphological processes and features
expected of its regime including
gradual aggradation,
channel widening,
the formation of depositional ridges
along the thalweg of the channel,
and the development of the depositional fan
(Fig. ).
SIMWE was the closest to the observed baseline
volumetric change.
It balanced erosion and deposition with
785.56 m3 of sediment eroded
and 608.91 m3 deposited
for a net change of -149.66 m3.
Only the SIMWE simulation deposited sediment
on the depositional fan.
While the difference of lidar surveys showed
that 244.60 m3 of sediment
were deposited on the fan,
SIMWE predicted that 54.05 m3
would be deposited.
SIMWE was unique in simulating unsteady flows
(Fig. a)
and fine-scale geomorphological processes
such as the development of depositional ridges
and a depositional fan.
While USPED generated a grainy pattern of erosion and deposition,
it was much faster than SIMWE
(Table )
and still simulated
the key morphological patterns and processes –
channel incision, filling, and widening.
Given their speed
and approximate modeling of erosive processes,
RUSLE3D and USPED
are effective for simulating landscape evolution
on large rasters.
RUSLE3D, for example, has been used to
model erosion for the entire 650 km2
Fort Bragg installation at 9 m resolution
.
Discussion
Limitations of this landscape evolution model include
shallow overland flow, units, computation time, and raster size.
r.sim.terrain only models shallow overland flows,
not fluvial processes or subsurface flows.
It requires data – including
elevation and rainfall intensity – in metric units.
The implementation of SIMWE in GRASS GIS
is computationally intensive
and may require long computation times even with multithreading.
Because SIMWE uses a Green function Monte Carlo solution
of the sediment transport equation,
the accuracy, detail, and smoothness of the results
depend on the number of random walkers.
While a large number of random walkers will reduce the
numerical error in the path sampling solution,
it will also greatly increase computation time.
A customized compilation of GRASS GIS
is needed to run SIMWE with more than 7 million random walkers.
This limits the size of rasters
that can be easily processed with SIMWE,
while RUSLE3D and USPED are much faster, computationally efficient,
and can easily be run on much larger rasters.
In the future, we plan to assess this model
by comparing simulations against
a monthly time series
of submeter-resolution surveys
by unmanned aerial systems and terrestrial lidar.
We also plan to develop a case study demonstrating
how the model can be used as a planning tool
for landscape restoration.
Planned enhancements to the model include
modeling subsurface flows,
accounting for bedrock,
and a reverse landscape evolution mode
for backward modeling.
Conclusions
The short-term landscape evolution model
r.sim.terrain can simulate the development of
gullies, rills, and hillslopes by overland water erosion
for a range of hydrologic and soil erosion regimes.
The model is novel for simulating landscape evolution
based on unsteady flows.
The landscape evolution model was tested
with a series of simulations for different
hydrologic and soil erosion regimes
for a highly eroded subwatershed on Fort Bragg
with an active gully.
For each regime, it generated the
morphological processes and features expected.
The physics-based SIMWE model
simulated morphological processes
for a variable erosion–deposition regime such as
gradual aggradation, channel widening,
scouring, the development of
depositional ridges along the thalweg,
and the growth of the depositional fan.
The empirical RUSLE3D model simulated channel incision
in a detachment-limited soil erosion regime,
while the semi-empirical USPED model
simulated channel widening and filling
in a transport-limited regime.
Since r.sim.terrain is a GIS-based model that simulates
fine-scale morphological processes and features,
it can easily and effectively be used
in conjunction with other GIS-based tools
for geomorphological research,
land management and conservation,
erosion control, and landscape restoration.
Code and data availability
As a work of open science,
this study is reproducible, repeatable, and recomputable.
Since the data, model, GIS, and dependencies are all
free and open source, the study can easily be reproduced.
The landscape evolution model
has been implemented in Python as a module
for GRASS GIS, a free and open-source GIS.
The source code for the model is hosted on GitHub at
https://github.com/baharmon/landscape_evolution (last access: 3 July 2019)
under the GNU General Public License version 2 .
The code repository also includes Python scripts
for running and reproducing the simulations in this paper.
The digital object identifier (DOI)
for the version of the software documented in this paper is
10.5281/zenodo.3243699.
There are detailed instructions for running this model in the manual at
https://grass.osgeo.org/grass76/manuals/addons/r.sim.terrain.html (last access: 3 July 2019)
and the tutorial at
https://github.com/baharmon/landscape_evolution/blob/master/tutorial.md (last access: 3 July 2019) .
The geospatial dataset for the study area is available on GitHub at
https://github.com/baharmon/landscape_evolution_dataset (last access: 3 July 2019)
under the
Open Database License (https://opendatacommons.org/licenses/odbl/, last access: 3 July 2019)
with the DOI:
10.5281/zenodo.3243700.
The
data log (https://github.com/baharmon/landscape_evolution_dataset/blob/master/nc_spm_evolution/DATA.md, last access: 3 July 2019) has a complete record of the commands used to process the sample data.
The source code, scripts, data, and results are also hosted
on the Open Science Framework at
https://osf.io/tf6yb/ (last access: 3 July 2019)
with the DOI
10.17605/osf.io/tf6yb.
Author contributions
BH developed
the models, code, data, case studies, and manuscript.
HM contributed to the development
of the models and case studies and revised the manuscript.
AP and VP
contributed to the development of the code.
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We acknowledge the GRASS GIS Development Community
for developing and maintaining GRASS GIS.
Review statement
This paper was edited by Bethanna Jackson and reviewed by three anonymous referees.
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