GEOPHYSICAL RESEARCH LETTERS, VOL. xx, Lxxxxx, doi:xx.xxxx/2009GLxxxxxx, 2009

--- Draft: 3/1/09 ---

Modeling a channel migration corridor for the 59-mile segment 
of the Missouri National Recreation River

Sky Coyote (Consultant to Oregon State University, Corvallis, Oregon, USA.),
Stephen T. Lancaster (Department of Geosciences, Oregon State University,
Corvallis, Oregon, USA.),
Robert B. Jacobson (US Geological Survey, Columbia, Missouri, USA)

Received xx March 2009; revised xx xxxxxxxx 2009; accepted xx xxxxx 2009;
published xx xxxxx 2009.

[1]
A novel procedure is presented to predict a 100 year migration corridor for the
59-mile segment of the Missouri National Recreational River from Gavins Point
Dam to Kensler's Bend (from Yankton, SD, to Ponca, NE). This prediction is based
on running multiple river simulations using the Johannesson-Parker 1989
meandering model. Computer programs were written in Python to digitize the banks
of the Missouri River from USGS topographic maps and Google Earth satellite
images, to calculate the river centerline and width, to digitize the river
valley boundaries, to perform the meandering simulations while maintaining river
points within the boundaries, and to check for and remove loop cutoffs. These
programs were used to run 3612 slightly different 100 year simulations which
accumulated 2d statistics showing the individual and aggregate locations of the
simulated rivers over time, and to perform an analysis of river evolution with
respect to input parameters. This data will be useful in planning river
management and land use during the next century.

Citation:

Coyote, S., S. T. Lancaster, and R. B. Jacobson (2009), Modeling a channel
migration corridor for the 59-mile segment of the Missouri National Recreation
River, Geophys. Res. Lett., xx, Lxxxxx, doi:xx.xxxx/2009GLxxxxxx.

1. Introduction

[2]
During the past 200 years, the shape and behavior of the Missouri River has been
changed considerably by human intervention.  While the Missouri River of 200
years ago was generally free-flowing and unpredictable, the river of today is
tamed by 600 miles of reservoirs, 800 miles of artificial channels, and six
dams. However, in 1978, 59 miles of the river were designated as a nationally
protected area where the conditions and environment would be maintained closer
to their ancestry.  In 1991, another 39 miles of the river were included in this
mandate, the two segments together being known as the Missouri National
Recreational River. The 59-mile segment of the MNRR is situated at the eastern
border between South Dakota and Nebraska, extending from Gavins Point Dam at the
eastern edge of Lewis and Clark Lake, south-eastward for approximately 95 river
kilometers to the city of Ponca, NE. This segment of the MNRR remains closer to
the shape, appearance, and behavior of 200 years ago, and is relatively free to
meander across the surrounding valley much as it did before human intervention
[NPCA website].

[3]
The objective of our work was to predict the 100 year migration of this segment
of the MNRR by simulating the evolution of its geometry using one or more
published meandering models.  However, based on comparisons of the actual river
geometry to that produced by simulation, we discovered that important aspects of
the behavior were not represented or predicted by the current 'state of the art'
models, especially the Johannesson-Parker 1989 model used in this study
[Johannesson and Parker, 1989].  This is primarily due to the fact that these
models represent the river as an idealized single channel having a centerline,
width, and a trapezoidal cross-section, and do not account for the multiple
channels and braids, or the numerous chutes, islands, and submerged and surface
bars which are features of the real river. To do this would require the
developent of a new model of river evolution which was beyond the scope of the
present work. Instead, we sought the simplest solution that captured most of the
physics of the river, but which made use of a large set of slightly different
simulations using the traditional methods to yield an aggregate probabilistic
result.  Thus, rather than attempt to predict the evolution of a single
simulated river with great accuracy, we decided to use a large set of
simulations having moderate accuracy to make statements about the likely
evolution of the real river based on the behavior of this population. The '100
year migration corridor' for the river was then to be determined from the
fraction of the population which reached various points within the river valley
during the course of the simulation, and from the boundaries that were defined
by these regions and the area they enclosed.

2. Materials and Methods

[4]
All computer software created for this study was written in the Python language,
also using the WxPython graphics library. Python is a high-level cross-platform
object-oriented programming language which is well known for its ease of use
and rapid development time. Software was developed and run under the Apple
Macintosh OS X 10.5 operating system, on an Apple iMac 2.4 GHz Intel Core-2 Duo
with 4 Gb of Random Access Memory and on an Apple MacBook 2.0 GHz Intel Core-2
Duo with 2 Gb of RAM.

[5]
A large mosaic was made from 24 1957-1996 USGS 7.5' topographic maps. These maps
were rasterized from PDF, rotated slightly, and aligned by hand in Adobe
Photoshop. The overall map projection was polyconic. The completed mosaic
contains sufficient detail for digitizing the river banks to better than 10 m
resolution. A simple Geographic Information System (GIS) program was written and
used to digitize the coordinates of the MNRR river banks and valley walls from
the map mosaic, and to interpolate these paths to a resolution of 100 m using
circular arcs. We did not include chutes or other bifurcations with the main
flow, but did included several islands. We also attempted to combine conflicting
bank data from maps made at different times. The valley boundary was digitized
along the 1200' topographic line, whereas the river falls from about 1170' at
Gavins Point Dam to about 1100' at Ponca. The meandering model requires a river
centerline and width at all points as input, and the GIS program computes these
automatically. Distances between points are calculated using oblate ellipsoid
geometry. A second mosaic was made from 17 Google Earth satellite images which
were also assembled by hand.

[6]
Another program was written to simulate and display the evolution of the 59-mile
segment of the MNRR. This program reads text files describing the shape of the
river, and name/value pairs of parameters which control the simulation.
Parameters can also be entered into the graphical user interface of a control
window, and results can be viewed in a plot window as the simulation is running.
The program also accumulates the cutoffs which occur as the river loops back
over itself. The meandering calculations are performed in a separate thread of
execution so that the user interface and plot animation does not block. The
meandering model used is that of Johannesson and Parker in their 1989 paper. The
JP89 model is based on several physical inputs, including the width and depth of
the river, flow volume, bed grain size, and water surface slope, as well as the
curvature at each point along the river. This model makes use of 2 coupled
integral equations of the down-stream distance. The curvature at each point is
calculated as the integral of 'local' curvature (defined as the inverse radius
of a circle fit to every 3 consecutive points) along a short interval of the
upstream river. The JP89 method calculates a linear speed perturbation at the
outside bank of each river bend which is based on a secondary cross-stream flow
which is perpendicular to the centerline. The migration rate is proportional to
this perturbation, which erodes the outside bank and deposits sediment on the
inside bank, but does not alter the width of the river.

[7]
The final program written was a 2-part database which accumulates information
about all simulations as a post-processing step, and displays and manipulates
these results in both numerical and graphical form. A database creation program
reads all completed simulation files and gathers several numerical results in 1-
and 2-dimensions, creating a large table of values. A second database query
program can perform numerical and logical operations on this table. The database
can track the course of every individual river over time, can extract specific
examples of rivers (such as the greatest or least lengths, coverage areas, or
erosion rates, or rivers which pass through specific points in the valley at
some time during the simulation), and can plot histograms and trends in the
behavior indicated by relationships between 2 or 3 input or output variables.

[8]
The JP89 meandering model does not generate the geometry and dynamics which are
specific to the MNRR. This model simulates a single channel with a trapezoidal
cross-section, while the MNRR contains many bifurcations, chutes, islands,
braids, and submerged and surface bars, making it an especially challenging
subject to model [Elliott and Jacobson, 2006]. We hypothesized that an
intelligent choice of many slightly different simulations, using a carefully
selected range of input values, would better approximate, as a population, the
likely behavior of the real river over the next 100 years than could any single
simulation. The ability to perform mutiple simulations was an enhancement of the
single-simulation program, which was modified to read and parse text files
containing the commands to run any number of different simulations. Based on
tests with individual combinations of the parameters used by the JP89 model, we
chose 7 different parameters as variable inputs to a large set of simulations:
the maximum attenuated width of the river, the depth, the flow volume, and bed
particle diameter (all constants), plus the 'lag distance' and 'sum distance'
which control the location and length (in river widths) of the curvature
integration, and the initial erosion rate. By examining the final configurations
of rivers generated from different combinations of these parameters, we chose
the following discrete values as inputs to the large set of runs:

Maximum width: 500, 600, 700, 800 m
Depth: 2.75, 3.0, 3.25, 3.5 m
Flow: 800, 1000, 1200, 1400 m^3/s
Particle diameter: 0.0005, 0.00075, 0.001, 0.00125 m
Lag distance: 1.0 only
Sum distance: 0.75, 1.0, 1.25 local widths
Initial erosion rate: 1.2, 1.8, 2.4, 3.6 ha/yr/km

[9]
Taken in all combinations, this yields 4x4x4x4x1x3x4 = 3072 total runs,
requiring an estimated 37.33 days to complete. We also decided to perform an
additional 540 simulations using a simpler meandering method which does not make
use of the perturbation in far bank speed, to compare and contrast with the
aggregate results of the JP89 model.

3. Results

[10]
Every simulation accumulated a 2-dimensional coverage and residence time array
showing how long the river stayed at any one place as it moved across the valley
during 100 years.  This array was saved every 10 years during the simulation,
and shows the cumulative amount of time that the simulated river occupied every
125 m x 125 m square in the valley.  The non-zero values of this array show the
maximum range of migration during this time, which includes cutoffs. These
arrays can be combined for all simulated rivers, or for subsets of rivers having
specific values or ranges of input parameters (such as 'flow'), to yield a
'common coverage array' for the set. The common coverage array can then be
plotted translucently on top of the map or satellite mosaic.  Contour lines of
this array show the range of motion of different fractions of the simulated
rivers, indicating how far from 1% to 99% of all rivers have traveled. The
common coverage array provides a probabilistic interpretation of the potential
migration of the actual river. The value of an element of this array indicates
the likelihood that the actual river will eventually reach that point at some
time during the next 100 years. The 'migration corridor' is then defined as the
subset of the common coverage array having values greater than or equal to some
constant, perhaps 0.95 for a 95% confidence level. This corridor will be
different for different confidence levels, and at different times during the 100
year simulation.  We believe that this provides a realistic and accurate picture
of what the actual 59-mile segment of the MNRR might do during the next 100
years.

[11]
Using the GIS program, 'target' areas can be defined within the valley which
correspond to towns, roads, airstrips, parks, and other geographic landmarks.
The database program can then compare these coordinates to the coverage area and
residence time arrays of each river simulation. Intersections of these two sets
indicate that part of a target has been reached by a particular river, and a
'probability of invasion' over the entire 100 year period can be calculated for
each target area by combining all rivers. Since the coverage arrays for each
simulation are saved at 10 year intervals, the database can also determine the
first time that any river reaches a specific target, if at all. These times can
be averaged together for all rivers which reach designated targets to yield the
probable time of arrival of the actual river. The database program can also find
examples of individual rivers which reach one or more designated targets at some
specific time during the simulation, but which miss other targets.

4. Discussion

[12]
The objective of this study was to develop a quantitative prediction of motion
of the 59-mile segment of the MNRR that delineates migration domains and ranges
which are amenable to potentially different management strategies. Although the
JP89 model was not sufficient to develop a prediction based upon a single
simulation of the river, it was able to do so when employed in a novel way to
generate a large population of simulations, each of which contributed a small
part to the overall prediction. Nevertheless, the present work suggests that the
development of a multiple-thread meandering model more appropriate to the
specific 3d geometric characteristics and diverse physical composition of the
MNRR should be pursued as a subsequent project.

[13]
Our work is based on the hope that an understanding of the variability of MNRR
geometry and dynamics can contribute important information to people who make
decisions about the management of the river and its resources. A quantitative
prediction of channel migration that delineates the probable limits, both
nominal and extreme, of river motion during the next century should allow
managers to assess proposed intervention activities based on the locations of
reaches in which migration rates are higher or lower than the norm. Such
information can be used to determine where channel migration can be tolerated,
and where it presents unacceptable risks or conflicts with other uses of the
river.

[14]
Additional information about this project can be found at
http://www.skycoyote.com/sky/MNRR/.

[15]
Acknowledgments. This research was supported in part by the National Park
Service.

References

Elliott, C. M., and R. B. Jacobson (2006), Geomorphic classification and
assessment of channel dynamics in the Missouri National Recreational River,
South Dakota and Nebraska, USGS Scientific Investigations Report 2006-5313.

Johannesson, H., and G. Parker (1989), Velocity redistribution in meandering
rivers, Journal of Hydraulic Engineering, Vol. 115, No. 8.

National Parks Conservation Association (website), Missouri National
Recreational River, http://www.npca.org/stateoftheparks/lewis_clark_trail/
mnrr.html.

S. Coyote, Coyote Consulting, http://www.skycoyote.com/sky/.

Stephen T. Lancaster (corresponding author), Department of Geosciences, Oregon
State University, Corvallis, OR, 97331, USA. (lancasts@geo.oregonstate.edu)

R. Jacobson, USGS Columbia Environmental Research Center, 4200 New Haven Road,
Columbia, MO  65201, USA.

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Figure 1. Examples of simulations of the 59-mile segment of the MNRR using the
JP89 meandering model: (a) the longest river (245.47 km); (b) the shortest river
including cutoffs (126.17 km); (c-d) nominal rivers within 0.1 standard
deviations of the average length (164.20 km) and erosion rate (2.05 ha/yr/km).
All plotted on a 1 km (small) and 10 km (large) square grid.

Figure 2. The 'common coverage array' of 3072 simulations plotted at 25%
opacity, plus contour lines showing the migration ranges of 99% (narrowest),
95%, 75%, 50%, 25%, 5%, and 1% (farthest) of all simulated rivers, overlaid on
the satellite image mosaic showing the locations of 13 'target' areas within the
MNRR valley.

Figure 3. The common coverage array for simulations having different values of
the input parameter 'flow'.  While the total coverage and coverage inside the
1-5% contours attains its maximum at 1400 m^3/s, coverage inside the 25-50% and
75-99% contours attains its maximum at 1200 m^3/s, and coverage inside the 75%
curve does so at 1000 m^3/s.

Table 1. Parameters used in large set of simulations.