U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigation Report 2008–5075

Prepared in cooperation with the

GRAND CANYON MONITORING AND RESEARCH CENTER

Modeling Water-Surface Elevations and Virtual Shorelines 

for the Colorado River in Grand Canyon, Arizona 
Cover Photo: Colorado River and the Unkar Delta from Lipan Point, Grand Canyon National Park, 

Arizona, by C.S. Magirl in 2003.
Modeling Water-Surface Elevations and 

Virtual Shorelines for the Colorado River in 

Grand Canyon, Arizona

By Christopher S. Magirl, Michael J. Breedlove, Robert H. Webb, and Peter G. 

Griffiths

Prepared in cooperation with the

GRAND CANYON MONITORING AND RESEARCH CENTER

Scientific Investigation Report 2008-5075

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

Robert D. Meyers, Director

U.S. Geological Survey, Reston, Virginia: 2008

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Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the 

U.S. Government.

Suggested citation:

Magirl. Christopher, F.N., Webb Robert, Griffiths Peter, 2008, Modeling Water-Surface Elevations and Virtual 

Shorelines for the Colorado River in Grand Canyon, Arizona, U.S.Geological Survey Scientific Investigations Report 

2008-5075, 32p
iii

Contents

Abstract ..........................................................................................................................................................1

Introduction.....................................................................................................................................................2

Grand Canyon Geomorphology ..........................................................................................................2

The STARS Model .................................................................................................................................2

Other Grand Canyon Hydraulic Models ............................................................................................3

Purpose and Scope ..............................................................................................................................3

Acknowledgements..............................................................................................................................3

Study Area, Place Names, and Units........................................ 4

Water-Surface Profile Analysis with the Standard Step Method .........................................................4

Hydraulic Model Construction.....................................................................................................................5

Model Parameters................................................................................................................................5

Topographic Data and Water-Surface Profile .................................................................................6

Cross Section Locations......................................................................................................................6

Generating Cross Sections .................................................................................................................6

Roughness Coefficient.......................................................................................................................10

Synthetic Bathymetry.........................................................................................................................11

Synthetic Bathymetry Calibration ....................................................................................................11

Evaluating Synthetic Bathymetry.....................................................................................................12

Estimating Model Accuracy.......................................................................................................................13

Virtual Shoreline Construction...................................................................................................................14

Results........................................................................................... 14

Analysis of Synthetic Bathymetry....................................................................................................14

Inundation Maps.................................................................................................................................19

Accuracy Estimate for flows under 1,400 m3/s...............................................................................19

Modeling Floods between 1,400 and 2,500 m3/s.............................................................................20

Modeling Large Floods.......................................................................................................................23

Suggested roughness values for flows below 1,400 m3/s ............................................................28

Future Model Adaptations..........................................................................................................................28

Conclusions...................................................................................................................................................29

References Cited..........................................................................................................................................30

Figures

Figure 1. Map of Grand Canyon showing corridor of the Colorado River and the 45 NAU monitoring sites and 27 Konieczki sites used to evaluate the stage-discharge prediction 

accuracy of the hydraulic model. ..............................................................................................1

Figure 2. Schematic diagram of a longitudinal profile defining terms used in one-dimensional 

hydraulics.......................................................................................................................................5

Figure 3. Cross-section locations shown for the reach of river at Sockdolager Rapid 

(river mile 79.1). ............................................................................................................................7

Figure 4. Cross section located at river mile 72.536 showing effect of trees on DEM 

topography.. ...................................................................................................................................8
iv

Figure 5. Sample section of river show downstream lengths for a cross section along the 

channel, along the left overbank, and along the right overbank. .........................................9

Figure 6. Schematic diagram showing definitions of terms in a channel cross section...............9

Figure 7. Example cross sections showing shape of synthetic bathymetry..................................11

Figure 8. The water-surface profile near Anasazi Bridge showing President Harding Rapids 

and several downstream rapids and results of the sensitivity analysis near Anasazi 

Bride showing the actual hydraulic radius measured by NAU and the hydraulic radius 

as predicted by the model calibrated for Manning’s n.........................................................17

Figure 9. Virtual shorelines at Eminence Break Camp (river mile 44.5)..........................................18

Figure 10. Comparison of the hydraulic model predictions and the stage-discharge data as 

measured by NAU at Eminence Break Camp near river mile 44.5. ....................................19

Figure 11. Comparison of the hydraulic model predictions and the stage-discharge data as 

measured by NAU at Dino Camp near river mile 50..............................................................19

Figure 12. Virtual shorelines at River Mile 50. The effects of the vegetated surface in the DEM 

can be seen as island polygons. ..............................................................................................20

Figure 13. The average of the residuals of the hydraulic model predictions compared against 

stage-discharge data measured at the NAU monitoring sites ...........................................21

Figure 14. Sensitivity analysis of the model residual mean (averaged for all NAU sites) for 

different values of Manning’s n................................................................................................21

Figure 15. The residual error calculated at six sites for discharges from 1,400 to 2,500 m3/s.....21

Figure 16. Matched photos showing the section of river at Boulder Narrows 

(river mile 18.746) .......................................................................................................................22

Figure 17. Hydraulic model cross section at river mile 18.746 (Boulder Narrows) showing 

predicted water-surface elevation at 227 m3/s (the shaded area) and the predicted high 

water mark of the 3,570 m3/s flood that occurred in Grand Canyon in 1957 .....................23

Figure 18. Virtual shorelines (shown as lines) and driftwood strands (shown as dots) for 

different floods at Palisades Creek (river mile 66) generated with a Manning’s n value 

of 0.035..........................................................................................................................................24

Figure 19. Reach of river in Granite Park (river mile 209) showing the broad boulder bar 

formed from reworked particles entering from the tributary on river left. The flow of the 

river in the image is from top to bottom. .................................................................................25

Figure 21. Virtual shorelines at Granite Park (river mile 209) as predicted by the hydraulic 

model. The line A-A’ extends between the 708 and 1,274 m3/s shorelines and rises 1.9 

meters across a distance of 100 meters.................................................................................26

Figure 22. The stage-discharge curve for the gaging station for Colorado River near Grand 

Canyon, Arizona (09402500) compared to the predicted stage-discharge curves from 

the hydraulic model for n=0.035, n=0.040, and n=0.045 (gage data taken from Topping 

and others, 2003). .......................................................................................................................27

Figure 23. The stage-discharge curve for the gaging station for Colorado River at Lees Ferry, 

Arizona (09380000) compared to the predicted stage-discharge curve from the 

hydraulic model for n=0.025, n=0.030, n=0.035, and n=0.040 ...............................................28

Figure 24. The average of the residuals of the hydraulic model predictions compared against 

stage-discharge data measured at the NAU monitoring sites when then Manning’s 

roughness coefficient has been tuned to produce optimal model results. ......................29
v

Tables

Table 1. Significant flows simulated in this study. ...........................................................................15

Table 2. Mean hydraulic radius in nine reaches as measured by Northern Arizona University 

and as predicted by the Grand Canyon hydraulic model.....................................................16

Table 3. Comparison of measured and model-predicted water velocity at low flow and high 

flow for the Colorado River in Grand Canyon.........................................................................16

Table 4. Change in mean hydraulic parameters for a discharge of 227 m3/s as a function of 

Manning’s roughness coefficient for the reach near Anasazi Bridge (river miles 42.669 

to 45.453). .....................................................................................................................................17

Table 5. Change in mean hydraulic parameters for a discharge of 227 m3/s as a function of 

Manning’s roughness coefficient for the reach of river in Inner Granite Gorge (river 

miles 86.567 to 87.946). ..............................................................................................................18

Table 6. Error measured relative to all NAU monitoring sites at varying discharge with a 

constant roughness coefficient of n=0.035. ...........................................................................20

Table 7. Error measured relative to all NAU monitoring sites for discharge up to 1,274 m3/s 

with the recommended roughness coefficients....................................................................28
Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Volume

cubic foot (ft3

) 0.02832 cubic meter (m3) 

cubic yard (yd3

) 0.7646 cubic meter (m3) 

Flow rate

cubic foot per second (ft3

/s) 0.02832 cubic meter per second (m3/s)

cubic foot per second per square

mile [(ft3

/s)/mi2]

 0.01093 cubic meter per second per square

kilometer [(m3/s)/km2]

cubic foot per day (ft3

/d) 0.02832 cubic meter per day (m3/d)

Vertical coordinate information is referenced in meters to height above the GRS80 ellipse 

defined by NAD83(1992). 

Horizontal coordinate information is referenced to North American Datum of 1983, NAD83(1992).

Elevation, as used in this report, refers to distance above the vertical datum.

Abbreviations and Acronyms

AML Arc Macro Language

DEM Digital Elevation Model

GCMRC Grand Canyon Monitoring and Research Center

GIS Geographic Information System

GUI Graphic User Interface

HEC Hydraulic Engineering Center

ISP Integrated Science Program

LIDAR Light Detection and Ranging

LOB Left overbank

NAU Northern Arizona University

RAS River Analysis System

ROB Right overbank

STARS Sediment Transport and River Simulation

USGS United States Geological Survey

WSPRO Water Surface Profile
Modeling Water-Surface Elevations and Virtual Shorelines 

for the Colorado River in Grand Canyon, Arizona

By Christopher S. Magirl, Michael J. Breedlove, Robert H. Webb, and Peter G. Griffiths

Abstract 

Using widely-available software intended for modeling rivers, a new one-dimensional hydraulic model was developed for 

the Colorado River through Grand Canyon from Lees Ferry to Diamond Creek. Solving one-dimensional equations of energy 

and continuity, the model predicts stage for a known steady-state discharge at specific locations, or cross sections, along the river 

corridor. This model uses 2,680 cross sections built with high-resolution digital topography of ground locations away from the 

river flowing at a discharge of 227 m3/s; synthetic bathymetry was created for topography submerged below the 227 m3/s water 

surface. The synthetic bathymetry was created by adjusting the water depth at each cross section up or down until the model’s 

predicted water-surface elevation closely matched a known water surface. This approach is unorthodox and offers a technique 

to construct one-dimensional hydraulic models of bedrock-controlled rivers where bathymetric data have not been collected. An 

analysis of this modeling approach shows that while effective in enabling a useful model, the synthetic bathymetry can differ 

from the actual bathymetry. The known water-surface profile was measured using elevation data collected in 2000 and 2002, 

and the model can simulate discharges up to 5,900 m3

/s. In addition to the hydraulic model, GIS-based techniques were used to 

estimate virtual shorelines and construct inundation maps. 

The error of the hydraulic model in predicting stage is within 0.4 m for discharges less than 1,300 m3/s. Between 

1,300-2,500 m3/s, the model accuracy is about 1.0 m, and for discharges between 2,500-5,900 m3/s, the model accuracy is on the 

order of 1.5 m. 

In the absence of large floods on the flow-regulated Colorado River in Grand Canyon, the new hydraulic model and the 

accompanying inundation maps are a useful resource for researchers interested in water depths, shorelines, and stage-discharge 

curves for flows within the river corridor with 2002 topographic conditions.

UTAH 

NEVADA 

Colorado

Rive

r

Kanab

Creek

Havasu

Creek

Lit t le

C o lorado

Paria River

Lake Powell 

Lake Mead 

Lees 

Ferry 

Gage 

Glen 

Canyon 

Dam 

Diamond Creek 

50 

100 

200 

250 

150 

Grand 

Canyon 

Gage 

0 50 km 

ARIZONA 

NAU Montoring Site 

USGS Gage 

50 River Mile 

Konieczki Site 

37° 

112° 111° 110° 

36° 

Figure 1. Map of Grand Canyon showing corridor of the Colorado River and the 45 NAU monitoring sites and 27 Konieczki sites used to 

evaluate the stage-discharge prediction accuracy of the hydraulic model.
2 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

Introduction

Prediction of water inundation and stage, or the height of 

the water surface, at a particular location is commonly needed 

by researchers studying various scientific questions along the 

Colorado River in Grand Canyon. The Grand Canyon Monitoring and Research Center’s (GCMRC) Integrated Science 

Program (ISP) often requires stage information for many 

specific locations and areas within Grand Canyon to address 

resource issues affected by operations of Glen Canyon Dam. 

To estimate the river stage, particularly for floods, Grand 

Canyon researchers rely on predictive equations (Wiele and 

Torizzo, 2003), reconstructed water-surface elevations from 

photographs (for example, Topping and others, 2003), stagedischarge relations from temporary or established gaging 

stations (Gauger, 1996; Konieczki and others, 1997), or stagedischarge relations measured at specific study sites (Hazel 

and others, 2007). Modeled estimates of river stage are also 

available for 386 km of the Colorado River below Glen Canyon Dam (Randle and Pemberton, 1987), but these predictions 

were based on limited bathymetric data and approximate topographic data. Driftwood, flood deposits, and high-water marks 

determined from vegetation bands also offer clues into depth 

of inundation of different flows (see, for example, Draut and 

others, 2005). Even where stage is known, the spatial extent 

of inundation along the river corridor for specific discharge is 

unknown. Numerical models are sometimes the only way to 

predict water depth at particular locations in Grand Canyon for 

variable discharges.

Grand Canyon Geomorphology

The Colorado River in Grand Canyon has long, flat 

sections of quiet water separated by steep, turbulent rapids. 

Periodic debris flows and frequent flash flooding originating 

in tributaries build debris fans at tributary mouths and deposit 

large boulders in the river (Cooley and others, 1977; Webb and 

others, 1989; Melis and others, 1994; Webb, 1996). In all, 534 

tributaries supply coarse-grained sediment into the Colorado 

River between Lees Ferry and Diamond Creek (Webb and 

other, 2000). The Colorado River, confined by bedrock walls, 

pools upstream of the accumulated debris fans before descending as rapids over the fans, commonly plunging into downstream pools and forming strong recirculation eddies below 

the rapids usually on both sides of the river. When viewed 

in profile, the water surface is stepped, termed by Leopold 

(1969) as a pool-and-rapid morphology. Viewed in plan, the 

distinct pattern of debris fans, rapids, pools, and recirculation 

eddies is readily apparent at most tributaries; Schmidt and 

Rubin (1995) called this morphology the fan-eddy complex 

and documented its influence on the overall geomorphic 

framework in Grand Canyon.

This longitudinal configuration of pools and rapids 

results from the dynamic interplay between the addition of 

coarse-grained alluvium from tributaries and the subsequent 

removal, or reworking, of that material by main-stem Colorado River floods (Kieffer, 1985; Webb and others, 1999). 

On average, about five debris flows occur each year in Grand 

Canyon (Griffiths and others, 2004) aggrading the river 

and causing the river profile to rise. Floods, in turn, act to 

erode this material and lower the water-surface profile. The 

total change in the water-surface profile near rapids can be 

more than a meter, occurring in a single storm or flood event 

(Magirl and others, 2005).

The STARS Model

Randle and Pemberton’s (1987) STARS model was 

developed to predict sand transport down the river during 

typical dam releases. With closure of Glen Canyon Dam in 

1963, the mean annual sediment supply at Lees Ferry of 57±3 

million metric tons (Topping and others, 2000) was mostly 

impounded behind the dam, and the retention of sand within 

the river corridor to preserve camping beaches and benefit 

native aquatic ecology became a salient management goal 

(U.S. Department of Interior, 1995). 

Written in Fortran and intended to be run on large 

mainframe computers, the STARS model was an important 

technical achievement at the time. The model was the first 

comprehensive hydraulic model built for the sizeable reach of 

the Colorado River between Lees Ferry and Diamond Creek 

(fig. 1). The STARS model was also applied to the Colorado 

River between Glen Canyon Dam and Lees Ferry using cross 

sections measured in 1990. The model used a constant Manning’s n value of 0.035 for all cross sections, though roughness 

coefficient is less important for rapid-influenced, bedrockcontrolled rivers than alluvial rivers owing to the dominant 

hydraulic control exerted by the rapids (O’Connor and Webb, 

1988; Webb and Jarrett, 2002). Randle and Pemberton (1987) 

forced their model to a subcritical regime (that is, allowing 

conditions up to but not exceeding critical flow) assuming that 

while the river may be critical at each rapid, no reach of the 

river has supercritical flow for any significant distance. This 

assumption proved appropriate as the model results qualitatively captured the behavior of the river.

There were, however, understandable limitations to the 

model. STARS used 708 cross sections for 362 km of river, of 

which only 199 cross sections represented actual topography 

and bathymetry from Grand Canyon, all measured in pools or 

low-velocity reaches of the river. The other 509 cross sections, 

located at major rapids, were “interpolated,” or constructed 

using aerial photography. Channel width and side slopes were 

estimated assuming a trapezoidal cross section. Bathymetry 

at each interpolated cross section was synthesized by adjusting the channel bottom up or down until the predicted watersurface profile closely matched a known water-surface profile 

measured along the river. This unique approach of creating 

synthetic bathymetry when real bathymetry is otherwise 
Introduction 3

unavailable is a powerful tool enabling the construction of 

hydraulic models in bedrock-controlled rivers; to the best of 

our knowledge, Randle and Pemberton (1987) and the Bureau 

of Reclamation (2001) are the only previous researchers to 

apply this approach. In 1987, the only available water-surface-profile data had been measured in 1923 (USGS, 1924). 

Because the water-surface profile in Grand Canyon is known 

to have changed significantly during the 20th century (Magirl 

and others, 2005), the STARS model better represented the 

state of the river in 1923, rather than the current, regulated 

river. Furthermore, due to the motivation to model sediment 

flux during smaller, dam-release flows and the lack of available topographic data, the height of each cross section in the 

STARS model extended above the river only far enough to 

accommodate a discharge up to 2,500 m3/s. Despite these limitations, many researchers still use the predicted water-surface 

profiles from the STARS model to estimate stage (Bureau of 

Reclamation, 1996; Walters and others, 2000).

Other Grand Canyon Hydraulic Models

Bennett (1993) built on the work of Randle and Pemberton (1987), constructing hydraulic models at a reach from 

river miles 59.3-87.6 (Grand Canyon reach) and another short 

reach between river miles 164.0-166.6 (National Canyon 

reach) to model the movement of sands and gravels. Bennett incorporated Randle and Pemberton cross sections into 

his model, then added 43% more cross sections to the Grand 

Canyon reach and 89% more cross sections to the National 

Canyon reach to improve resolution. O’Connor and others 

(1994) constructed a small hydraulic model for a 1.2 km reach 

of the river near river mile 2.0 to estimate peak discharge of 

the paleofloods which left slackwater deposits at Axehandle 

Alcove. 

An unsteady, one-dimensional routing model was developed for Grand Canyon. This model solved the St. Venant 

equations using reach-averaged normalized cross sections and 

predicted the routing of daily dam-released discharge waves 

down the river corridor (Wiele and Smith, 1996; Wiele and 

Griffin, 1997). This model was latter enhanced to simulate the 

downstream transport of fine-grained sediment in the reach 

from Lees Ferry to Phantom Ranch at river mile 87 (Wiele 

and others, 2007). Though this unsteady model is useful for 

predicting the propagation of a discharge wave downstream, it 

cannot directly predict stage or water inundation at a particular 

site. 

Two-dimensional models have also been built for short 

reaches of the Colorado River in Grand Canyon. One such set 

of models was built for four sections of the Colorado River 

below the confluence with the Little Colorado River using a 

finite-volume approach (Wiele and others, 1996; Wiele, 1997). 

These Wiele models not only predicted stage and flow velocity, but also predicted the spatial aggradation or erosion of 

sand from the riverbed during flood conditions. Another twodimensional model was built to simulate flow near Mohawk 

Canyon (river mile 172) in Grand Canyon in order to predict 

water surface and flow velocity (Miller and Cluer, 1998). Due 

to the computational complexity of running these two-dimensional models, however, only short reaches can be simulated 

at a time and a simulation of all of Grand Canyon would be 

impractical. 

Purpose and Scope

With widely-available, GUI-based hydraulic software 

packages and an extensive, recently-collected Grand Canyon 

topographic data set (including LIDAR data from 2000 and 

photogrammetric DEMs from 2002), it became appropriate in 

the early 2000s to develop a new hydraulic model analogous 

to the STARS model, excluding sediment transport. This new 

model, incorporated into the HEC-RAS modeling system, uses 

one-dimensional, standard-step modeling to predict water-surface elevations along the main channel of the Colorado River 

between Lees Ferry (river mile -0.001) and the gaging station 

above Diamond Creek (river mile 225.228). In addition to 

developing an updated hydraulic model for Grand Canyon, we 

also developed GIS-based techniques to visualize and analyze 

the extent of water inundation at different discharges for any 

site in Grand Canyon. 

The purpose of this report is to document the creation and 

use of the one-dimensional hydraulic model and the accompanying graphics package depicting virtual shorelines in Grand 

Canyon. We built the hydraulic model to simulate steady-flow 

discharges on the Colorado River up to 5,900 m3

/s. Estimates 

of the accuracy of the model are given.

Acknowledgements

Development of the updated STARS model was funded 

by the Grand Canyon Monitoring and Research Center of the 

U.S. Geological Survey, Flagstaff, Arizona. Tim Randle supplied bathymetric data and provided suggestions and insight 

for the construction of the model. Joe Hazel provided the latest 

stage-discharge data from Grand Canyon monitoring sites. 

Steve Young took the photograph of Boulder Narrows at low 

flow. David Topping provided stage-discharge data from high 

flows at Lees Ferry and Grand Canyon and detailed driftwood 

data at multiple Grand Canyon sites. Tim Randle, David 

Schoellhamer, David Topping, and Mike Nolan reviewed 

the manuscript, with David Topping providing particularly 

insightful comments to evaluate the validity of the synthetic 

bathymetry approach and to improve the overall report. 

Finally, Helen Fairley was instrumental in motivating and supporting the final release of this report.
4 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

Study Area, Place Names, and Units

In this report, we use metric units for all measures except 

river mile, which is used to describe the location along the 

river centerline in Grand Canyon. Though a number of rivermile conventions are available, we used the standard established by GCMRC (2002). Use of river mile has considerable 

historical precedent and provides a reproducible method of 

describing sites with respect to the Colorado River. The left 

and right sides of the Colorado River are determined as one 

faces downstream. Similarly, consistent with common practice, cross sections are displayed and described as if looking 

downstream.

We typically refer to “Grand Canyon” in broad reference to the Colorado River watershed between Lees Ferry and 

the Grand Wash Cliffs, including Marble and Grand Canyon 

proper. “Marble Canyon” is the canyon reach of the Colorado 

River between Lees Ferry and the confluence with the Little 

Colorado River (river miles 0 to 61.5; fig. 1). “Glen Canyon” 

is the reach of the Colorado River upstream from Lees Ferry, 

most of which has been dammed by Glen Canyon Dam (river 

mile -15.4). The sub reach, “Upper Granite Gorge” is the 

section of river between river mile 77.5 and river mile 117.8, 

characterized by steep, narrow walls of Precambrian bedrock 

(Schmidt and Graf, 1990). The sub reach, “Furnace Flats” is 

the section of river between river mile 61.6 and river mile 77.4 

which is relatively wide, both in the channel and the overbanks (Schmidt and Graf, 1990).

Water-Surface Profile Analysis with 

the Standard Step Method

One-dimensional hydraulic modeling is a common technique to predict the depth of flow in natural and man-made 

channels. In building a model, flow is assumed to be steady 

and one-dimensional, gradually varied, and cross sections are 

constructed orthogonally to the flowing water to best represent 

the geometry of the river channel (Hoggan, 1997). The water 

surface spanning a cross section is assumed to be level, and 

the slope of the channel is assumed to be small. Cross sections 

are commonly generated using field surveys, digital elevation models (DEMs), or topographic maps, and in addition 

to including geometry (both above and below the existing 

water line), an estimate of the hydraulic roughness at the cross 

section is required. A full list of assumptions and techniques 

for one-dimensional modeling is available in Chow (1959), 

Davidian (1984), and Hoggan (1997).

The standard-step method is commonly used to estimate 

water-surface profiles in natural channels (Chow, 1959; Henderson, 1966). The method computes changes in water-surface 

elevation between adjacent cross sections by solving the 

energy equation written in the following form (fig. 2):

Y Z V

g

Y Z

V

g

h i i

i i

i i

i i + + = + + + + + e

a + + a 2

1 1

1 1

2

2 2 , (1)

where Y is the depth of the water at a given cross section, i is 

the index for the given cross section, i+1 is the index of the 

next upstream cross section (when assuming subcritical flow 

conditions), Z is the elevation of the main channel inverts, a

is the velocity weighting coefficient, V is the average velocity 

at a cross section, g is the gravitational acceleration, and he is 

the energy head loss between the two cross sections. To solve 

equation (1), the continuity equation at each cross section is 

needed to calculate average velocity:

 V Q

Ai

= , (2)

where Q is the discharge in the river and A is the cross sectional area of the flowing water. If the water surface is known 

at the downstream cross section (fig. 2), then combining 

equation (2) with equation (1) leaves two unknown variables, 

the water surface at the upper cross section and the energy 

loss between the cross sections. To reduce the number of 

unknowns downs to one, information of the energy losses is 

needed. Energy loss in open channels has two components,

 h h h e f = + o , (3)

the energy loss due to boundary roughness or friction, hf, 

and other (expansion and contraction) energy losses, ho. The 

energy loss due to boundary roughness is approximated by,

h S L f f = , (4)

where Sf is the representative mean friction slope and L is the 

reach length between the cross sections. The friction slope, 

and subsequently the energy loss due to boundary roughness, is calculated using Manning’s equation, an estimate 

of Manning’s roughness coefficient (n), as well as values of 

conveyance calculated for different regions of the channel. 

The conveyance calculations are also used to estimate the 

velocity weighting coefficient used in equation (1). Details of 

the Manning’s and conveyance equations and their application 

to the standard step method are not needed for the discussion 

in this report, but they can be found in Chow (1959) and Hoggan (1997). The other energy losses, ho

, appearing in equation 

(3) are due to expansion and contraction losses arising from 

eddies in the flow,
Hydraulic Model Construction 5

h C

V

g

V

g o = - i i + + i i

a 1 1 a

2 2

2 2 , (5)

where C is either the expansion coefficient (if the downstream 

velocity is less than the upstream velocity) or the contraction 

coefficient (if the downstream velocity is greater than the 

upstream velocity).

Equations (3-5) are used to solve the energy equation, 

but due to irregularities in natural channels, the upstream 

water surface cannot be explicitly solved, and an iterative set 

of steps is needed to determine the upstream water-surface 

elevation. First the water surface at the upstream cross section 

is assumed, and the velocity head and total conveyance at the 

upstream cross section is computed. With these computed 

values of velocity head and conveyance, the total energy loss, 

he, is computed with equation (3). The upstream water-surface 

elevation can then be calculated using the energy equation (1), 

and this water-surface elevation is compared to the water-surface elevation computed with the original assumption. If the 

difference between these two values is less than the accepted 

tolerance, the iteration is complete. If the difference between 

the two water-surface values is greater than the allowable tolerance, the steps above are repeated using the energy-equation 

computed water surface as the new assumed estimate. This 

iterative method is the standard-step method and is discussed 

in detail by Chow (1959) and Hoggan (1997).

Standard-step analysis is usually performed using a 

numerical model, and common computer programs for step 

analysis include HEC-2 (Feldman, 1981), WSPRO (Shearman 

and others, 1986), and HEC-RAS (Brunner, 2002). The differences among these programs are mainly the user interface and 

specific routines to handle flow around floodplain structures. 

Hydraulic Model Construction

The new Grand Canyon hydraulic model was constructed in HEC-RAS version 3.1, a free software application 

developed by the US Army Corps of Engineers, Hydrologic 

Engineering Center (Brunner, 2002). With an integrated 

graphical user interface, wide acceptance, and availability of 

the application, HEC-RAS proved to be an appropriate development platform. The model is one dimensional and uses the 

standard-step approach to predict the water-surface elevation, 

or stage, at specific cross sections. After development of the 

Grand Canyon hydraulic model, the Hydrologic Engineering 

Center released HEC-RAS version 4.0 beta; the Grand Canyon 

hydraulic model is compatible with this newer HEC-RAS 

version.

Model Parameters

The model was run under subcritical flow conditions. 

Supercritical flow is not sustainable for significant distances in 

most natural channels (Jarrett, 1984; Trieste, 1992). For rivers 

flowing over an alluvial substrate in a natural channel, many 

postulate that supercritical flow occurs rarely, if ever (Jarrett, 

1984; Grant, 1997). For this reason, it is common for researchers to run hydraulic models under subcritical conditions 

(Randle and Pemberton, 1987; O’Connor and Webb, 1988; 

Webb and Jarrett, 2002), where flow is allowed to approach, 

but not exceed critical depth. In Grand Canyon, where debris 

fans constrict the flow, HEC-RAS tends to predict critical 

flow at most significant constrictions, particularly for lower 

discharges.

The downstream cross section of the model was placed 

at the site of the USGS gaging station at Diamond Creek 

(Colorado River above Diamond Creek near Peach Springs, 

Arizona, 09404200) located at river mile 225.228. The 

downstream boundary condition was selected to be the rating 

curve from the gaging station. This Diamond Creek rating 

curve, however, is only valid for discharges up to 1,400 m3

/s. 

In order to model larger discharges, the shape of the adjusted 

rating curve from the USGS gaging station at Grand Canyon 

(Colorado River near Grand Canyon, Arizona, 09402500) was 

superimposed on the upper end of the Diamond Creek rating 

curve for discharges between 1,400 m3

/s to 5,900 m3/s. The 

geomorphology of the river corridor near the Diamond Creek 

and Grand Canyon gages is similar—narrow canyon walls of 

Precambrian granite and gneiss—and lacking high discharge 

flow data measured directly at the Diamond Creek gaging station, this substitution represented a reasonable surrogate.

 The contraction coefficient was kept at the default 

value of 0.1 and the expansion coefficient was kept at the 

default value of 0.3, though larger values may be appropriate for rapids reflecting abrupt transitions to better estimate 

the energy losses associated with recirculation eddies (Hoggan, 1997). Future versions of the model could benefit from 

analysis and application of more appropriate loss coefficients. 

#ROSS

3ECTION

 #ROSS

3ECTION

#ROSS 

3ECTION



7ATER
3URFACE

3LOPE

&RICTION

3LOPE

6ELOCITY

6

%NERGY'RADE,INE

7ATER
3URFACE0ROFILE

#HANNEL"ED

Figure 2. Schematic diagram of a longitudinal profile defining 

terms used in one-dimensional hydraulics.
6 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

Similarly, in modeling rapids with large recirculation eddies 

either upstream or downstream of the rapid, the use of ineffective flow areas may improve model results. HEC-RAS 

allows these ineffective flow areas to be built into cross sections within models to restrict the total area in a cross section 

available for flow conveyance. By inserting ineffective flow 

areas into cross sections dominated by eddies, future versions 

of the model may better predict stage, particularly in sections 

immediately downstream of rapids.

Topographic Data and Water-Surface Profile

Topographic data used to construct the subaerial sections 

of the model came primarily from a data set of digital imagery and automated photogrammetry provided to GCMRC by 

ISTAR America, Inc. ISTAR America collected and processed 

images captured during a series of May 2002 Grand Canyon overflights. In addition to digital imagery, a one-meter 

digital elevation model (DEM) built with photogrammetry 

(NGVD88) was assembled for the entire river corridor. The 

ISTAR DEM has an absolute vertical accuracy estimated by 

GCMRC to be better than 0.3 m (Steve Mietz, GCMRC, written communication, 2003). ISTAR aerial photography was 

also used to facilitate cross-section selection.

The water-surface profile used to calibrate the model was 

modified from a water-surface profile constructed by Magirl 

and others (2005); Magirl and others used LIDAR overflight 

data collected in 2000 to construct their original profile. 

Because the hydraulic model was built from topographic 

data from the ISTAR America DEM, this 2000 water-surface 

profile was mapped onto the ISTAR America DEM and points 

were adjusted vertically such that the spatial coverage of the 

water surface closely matched the extent of the river shoreline 

shown in the DEM and matching imagery. Typical adjustments 

were less than 0.5 m. The discharge for the water surface 

reported by Magirl and others (2005) was 227 m3

/s, as was the 

discharge during the 2002 overflights used to build the DEM. 

This adjustment process produced a water-surface profile 

matching the morphology of the DEM and represents the state 

of the river in 2002. This 2002 water-surface profile, used for 

model calibration, is available in Magirl (2006). 

While no independent estimate of the accuracy of the 

water-surface profile is available, it is possible that either 

the original LIDAR profile or the process of mapping this 

profile onto the DEM differs from the actual water-surface 

profile in the river. If there is a significant difference between 

the water-surface profile used to calibrate the model and the 

water-surface profile in the actual river corridor, errors could 

be introduced during the calibration process. This possibility 

will be discussed in the Synthetic Bathymetry section below.

Cross Section Locations

The locations of cross sections were chosen by following, 

when possible, the techniques recommended in the hydraulic 

literature (Benson and Dalrymple, 1967; Davidian, 1984). 

Ideally, cross sections would be chosen away from eddies 

where the entire flow field is downstream and perpendicular 

to the cross section. Owing to the ubiquitous distribution of 

eddies along the river corridor in Grand Canyon, however, it 

would be impractical when selecting cross sections to avoid 

all eddies. Many cross sections, therefore, are positioned 

across recirculation eddies. At each rapid, using the approach 

of Randle and Pemberton (1987), a group of at least four 

cross sections were selected: one at the head of the rapid, a 

second about 15 m upstream from the head, a third about 45 

m upstream from the head, and a fourth cross section at the 

foot of the rapid. For long rapids, additional cross sections 

were placed within the rapid as needed to capture the fall of 

the water surface moving down the rapid (fig. 3). A collection of four cross sections were also placed near tributaries 

that flowed into otherwise slow-moving reaches of river. 

These tributaries, while associated with no rapid today, were 

assumed to have the potential to create rapids in the future. 

Placement of extra cross sections near currently slow-moving 

tributary junctures allows adaptation to future modeling of 

tributary input; the model can also be quickly updated in the 

event of a rapid-creating debris flow. In slower sections of 

river away from tributaries, cross sections were spaced to best 

capture the progression of the river following the recommendations of Davidian (1984), and cross sections were placed 

at river sections that widened or constricted abruptly (expansions and constrictions). To enable a comparison of the model 

results with the STARS model, cross sections were also generated at the locations of the 199 non-interpolated cross sections 

built into the STARS model.

Generating Cross Sections

A total of 2,680 cross sections were generated and built 

into the Grand Canyon hydraulic model covering 362 km of 

river from Lees Ferry to Diamond Creek. Consistent with 

HEC-RAS geometric conventions, each cross section was 

generated as a two-dimensional list of coordinates representing the station/elevation values of points along the cross section when viewed in the downstream direction. Points within 

a cross section were spaced, on average, about 3 m apart. The 

bank station for the right and left banks (that is, those points in 

the cross section separating the main channel from the overbank regions) was placed where the cross section intersected 

the river shoreline at 227 m3

/s. We determined this location 

using aerial imagery. The cross sections were constructed 

perpendicular to the river centerline and extended away from 

the river shorelines, up the canyon walls, far enough to incorporate 30 m vertical elevation above the water surface on both 

sides of the cross section. This freeboard enabled the model 
Hydraulic Model Construction 7

to simulate discharge up to 5,900 m3/s at every cross section. 

Moreover, in HEC-RAS, if the depth of the predicted water 

surface rises above the extent of the available cross-section 

geometry, the model assumes that walls rise vertically from 

the edge of the cross section containing the flow. Therefore, 

discharges higher than 5,900 m3/s can be simulated along specific reaches where the predicted water surface overtops the 

existing cross sections. 

In some wider sections of the river where river curvature 

was pronounced, distal ends of adjacent cross sections occasionally crossed over each other. While cross sections would 

be ideally broken at one or more points to maintain the section 

roughly perpendicular to the flow (Benson and Dalrymple, 

1967), it was impractical to follow this labor intensive exercise 

when building the Grand Canyon model. Instead, we accepted 

that some sections may cross over each other and potentially 

affect the accuracy of the model. Future versions of the model 

may improve performance by reconstructing cross sections to 

avoid this interference. 

In places, the DEM contains topographic surfaces that 

incorporate the tops of heavy stands of riparian vegetation, 

overestimating the true elevation of the ground. Each cross 

section was therefore analyzed while juxtaposed against imagery to determine areas of vegetation-influenced topography. 

Wherever high points in the cross section were identified with 

known stands of riparian vegetation, the cross section was 

edited to remove these higher elevation points (fig. 4). The 

topography under the removed vegetation was assumed to be 

linearly consistent with the grade away from the vegetation. 

At other places within the DEM, depressions existed such 

that in some cross sections, DEM-obtained elevation points 

away from the river were below the known water-surface 

elevation for that cross section. These elevation artifacts were 

removed by manually setting the elevation at these depressed 

locations to the known water-surface elevation.

Within the model for a given cross section, the distances 

to the next downstream cross section along the centerline, 

along the left bank, and along the right bank are required 

(fig. 5). The downstream distance between cross sections 

along the centerline, D, is determined by location of the cross 

sections relative to the river centerline. To calculate downstream distance of each overbank region, the width of the 

river, w, the curvature of the river for the current cross section, 

θ1

, and the curvature of the river for the next downstream cross 

section, θ2, are required. The curvature is the angle between 

the downstream flow vectors of sequential locations along the 

river; counter clockwise is positive. Once D is determined, the 

overbank distances are calculated using the following equations:





















  METERS ±

&LOW

Figure 3. Cross-section locations shown for the reach of river at Sockdolager Rapid (river mile 79.1). The river-mile location appears 

next to each cross section. The length of each cross section represents the distance the cross section extends up the canyon walls.
8 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

810

775

780

785

790

795

800

805

0 5 0 100 150 200 250 300 350 400 450

R a w c r o s s s e c tio n

A ft e r t r e e r em o v a l

2-

E le v a t io n (m )

C r o s s S e c t io n S t a t io n (m )

&LOW

Figure 4. Cross section located at river mile 72.536 showing effect of trees on DEM topography. The top image is an aerial photo from 

ISTAR imagery data set; the white line represents the location of the cross section. The bottom graph shows the cross-section data 

(looking downstream) before and after the tree canopy was manually removed from the cross section.
Hydraulic Model Construction 9

&LOW

4ARGET83

$CH

$,/"

$2/"





W

Figure 5. Sample section of river show downstream lengths for a cross section along the channel, along the left overbank, and along 

the right overbank.

4HALWEG

,EFT/VERBANK 2IGHT/VERBANK

N,/" NCHANNEL N2/"

#HANNEL

,EFT"ANK3TATION 2IGHT"ANK3TATION

Figure 6. Schematic diagram showing definitions of terms in a channel cross section. The cross section is viewed in the downstream 

direction. The roughness coefficient, Manning’s n, can be specified uniquely for the channel, the left overbank region, and the right 

overbank region.
10 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

D D w

LOB c = -h +

2 1 2 (tan t q q an ) (6)

and

D D w

ROB c = +h +

2 1 2 (tan t q q an ) (7)

Roughness Coefficient

For each cross section, HEC-RAS requires values of 

Manning’s n to be specified within the channel, left overbank, 

and right overbank (fig. 6). However, the hydraulic model for 

Grand Canyon developed for this study is largely controlled 

by rapids located throughout the river corridor. These rapids 

act as hydraulic controls. Wherever flow is constricted to 

critical conditions in the channel, the water-surface elevation 

at the constriction, or the rapid, is determined by the critical depth at the constriction and not roughness coefficient. 

Roughness coefficient does become important, however, in 

determining water depth in subcritical reaches upstream from 

rapids. The farther upstream from a constriction, the greater 

the influence roughness coefficient can have on the predicted 

water-surface elevation. Away from cross sections with critical 

flow, Manning’s roughness coefficient, n, is an all inclusive 

energy consumption parameter that dictates the quantity of 

energy head loss between cross sections through equation (1). 

In slow-moving alluvial rivers, energy losses are manifested 

through, among other things, boundary friction and sediment 

transport. In steep-gradient rivers, or within rapids, surface 

waves, extreme turbulence, and recirculation eddies consume 

large amounts of energy, strongly affecting the appropriate 

choice of Manning’s n needed to accurately model the flow. 

All of these factors make the accurate selection of Manning’s 

n a difficult exercise for the Colorado River in Grand Canyon. 

In order to build the hydraulic model, an estimate of the 

roughness coefficient in the channel was needed to create 

the synthetic bathymetry (discussed below); the creation of 

the synthetic bathymetry was labor intensive, requiring over 

12 months to finish. After completing the hydraulic model, a 

series of analyses were then done to evaluate how accurately 

the synthetic bathymetry represented real bathymetry. These 

analyses suggested how, if the model were to be reconstructed 

or if this technique were to be used in another river system, a 

more accurate model could be created. In this study, however, 

the labor intensive task of model construction precluded a 

recalibration of the model using new roughness coefficients 

suggested by the analyses. In other words, the model was constructed only once, and we did not have the time to rebuild the 

model using information from the post-construction analyses.

For the model construction and synthetic bathymetry 

calibration, a standard Manning’s n roughness value of 0.035 

was chosen for all channel regions during the calibration of 

the synthetic bathymetry (see below). The bed of the Colorado 

River in Grand Canyon is a mixed substrate ranging from sand 

with various bed forms in slow reaches of the river to cobbles 

and boulders where flow velocity is high or where tributaries enter the river corridor (Howard and Dolan, 1981). Just as 

bathymetry is generally unavailable in most reaches of river, 

the substrate for any given cross section is generally unknown. 

The roughness value (n=0.035) chosen for the channel represented a best guess synthesis of all the roughness elements of 

the Colorado River in Grand Canyon and is consistent with 

the recommendations of Chow (1959). This is also the same 

value of roughness used by Randle and Pemberton (1987) in 

calibrating the synthetic bathymetry of the STARS model.

While the roughness in overbank regions is variable, a 

uniform Manning’s n value of 0.035 was also initially applied 

to all overbank regions. The overbank region of some cross 

sections spans exposed bedrock or talus slopes with large 

particles and little vegetation. Other cross sections have dense 

vegetation. Still other cross sections have vegetation close to 

the shoreline with open, low-roughness sections away from 

the shore. This initially chosen Manning’s n value of 0.035 

in the overbank area may not represent the actual value of 

roughness, but specific information for each of the 2,680 cross 

sections was not available. Thus, lacking better information 

to justify something different, the same Manning’s n value 

(n=0.035) of the channel was applied to the overbank regions. 

After completing the hydraulic model, error analysis of the 

results indicated that adjusted roughness values improved 

model accuracy for discharges below 1,400 m3

/s (see Results 

section). A future potential model improvement would be to 

more accurately estimate the roughness coefficients of overbank regions for each cross section.

While hydraulic roughness is held constant with increasing discharge in the Grand Canyon hydraulic model, it is recognized that hydraulic roughness varies with flow conditions 

(Rouse, 1965). In steep-gradient rivers, there is evidence that 

hydraulic roughness decreases with increasing water depth 

as the size of the roughness elements on the riverbed become 

smaller relative to the depth of water (Jarrett, 1984). Indeed, 

many roughness equations predict hydraulic roughness varying inversely with water depth (Limerinos, 1970; Jarrett, 1984; 

Bathurst, 2002; Yen, 2002). In contrast, Kidson and others 

(2006) showed roughness increasing as rising stage exits the 

channel and enters riparian vegetation on the floodplain of the 

Mae Chaem River of Northern Thailand. To examine the possibility of variable roughness values impacting the hydraulic 

model for Grand Canyon, particularly under flood conditions, 

a sensitivity analysis of the final model (that is, after building 

the synthetic bathymetry) with different Manning’s n values 

was performed. The model was run at discharges up to 1,274 

m3

/s with roughness coefficients ranging from 0.025 to 0.045. 

The model’s predicted stage-discharge behavior was then 

compared to the known stage-discharge curves at different 

sites along the river.
Hydraulic Model Construction 11

Synthetic Bathymetry

The cross sections built from the DEM contain only 

subaerial topography away from the river and no bathymetric 

data below the water surface. Synthetic bathymetry, therefore, 

was created for each cross section, mimicking the approach of 

Randle and Pemberton (1987). The water depth for each cross 

section was initially set to a value based on a bathymetric trace 

collected by Wilson (1986). This initial bathymetry was somewhat arbitrary and ultimately irrelevant as the bathymetric 

depths were later reset by the calibration process. The shape 

of the bathymetry at each cross section was assumed to be one 

of four styles based on the proximity of the cross section to 

tributaries (fig. 7). 

For cross sections adjacent to a tributary, a triangular 

shape was used to best represent the subaqueous wedge of 

coarse-grained alluvium debouched from the tributary. For 

example, if the cross section was adjacent to a tributary that 

entered from river left, a triangular shape was chosen with the 

thalweg located four-fifths of the total channel width away 

from the left bank (fig. 7b). Similarly, for cross sections near 

tributaries that entered from river right, a triangular shape was 

chosen with the thalweg located four-fifths of the total channel 

width from the right bank (fig. 7c). For cross sections with 

tributaries entering from both sides, a triangular shape was 

chosen with the thalweg placed in the channel center (fig. 7d). 

For cross sections away from any tributary, the bathymetry 

was assumed to be trapezoidal with 1:1 side slopes (fig. 7a). 

Similarly, for the few cross sections that had an island and two 

separate flow channels, the bathymetry for each channel was 

assumed trapezoidal with 1:1 side slopes. These cross-section 

shapes are consistent with the trends of channel shape in 

Grand Canyon reported by Flynn and Hornewer (2003).

Occasionally, cross sections showed small exposed 

boulders in the middle of the river. These particles were, for 

the most part, removed from the cross section. An exception 

to this rule was the large boulder at river mile 18.746, several 

meters wide in the middle of the channel. This large boulder 

was retained in the model as an island.

Synthetic Bathymetry Calibration

The goal, or intent, of the calibration process was to 

adjust the depths of cross sections such that the model’s 

predicted water-surface elevation matched the known watersurface elevation to within a predetermined tolerance limit. 

During the model calibration, the target tolerance between the 

known and predicted water-surface profiles was 30 mm and 

the calibration discharge was 227 m3/s. All synthetic-bathymetry calibration was done with a constant Manning’s roughness 

coefficient of n=0.035. Owing to the subcritical nature of the 

hydraulic model, calibration started at the farthest downstream 

cross section, working upstream until the entire model was 

calibrated. When changing bathymetry, station/elevation 

points within that cross section below the known water-surface 

elevation were adjusted vertically; the adjustment at a particuFigure 7. Example cross sections showing shape of synthetic bathymetry in (a) an open section of the river, (b) a section of river with 

tributary entering from river left, (c) a section of river with tributary entering from right, and (d) a section of river with tributaries entering 

both sides.

-5 0 0 5 0 100 150 200 250

955

910

915

920

925

930

935

950

945

940

E le v a t io n (m )

(a )

-5 0 0 5 0 100 150 200 250

(d )

R M 6 .6 0 6

R M 4 7 .7 9 3

870

830

835

840

845

850

865

860

855

875

C r o s s S e c t io n S t a t io n (m )

-5 0 0 5 0 100 150 200 250

(c )

R M 2 .7 3 7

955

915

920

925

930

935

950

945

940

960

C r o s s S e c t io n S t a t io n (m )

955

910

915

920

925

930

935

950

945

940

E le v a t io n (m )

(b )

-5 0 0 5 0 100 150 200 250

R M 4 .0 2 6
12 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

lar station/elevation point was made proportionately according to the relative depth of the point so that the shape of the 

bathymetry at that cross section remained consistent. Adjustments that would result in a water depth less than 1.52 m were 

not made; in other words, the minimum allowable water depth 

for a cross section was 1.52 m. This minimum depth was 

chosen arbitrarily.

The calibration process began by inspecting the difference between the predicted and known water-surface elevations at the target cross section. The synthetic bathymetry 

was then raised or lowered at the target cross section and its 

closest downstream neighbors (the downstream neighbors 

also needing adjustment due to the hydraulic influence from 

downstream control in subcritical flow). After rerunning 

the hydraulic model, a new predicted water-surface profile 

was again compared to the known profile, thus suggesting a 

change value for the next adjustment. By iteratively evaluating water-surface elevation difference, adjusting bathymetry, 

then reevaluating the difference, the predicted water-surface 

elevation at the target cross section eventually approached the 

known water surface. If the difference between the predicted 

and known water-surface elevation was less than the accepted 

tolerance, the next upstream cross section became the new 

calibration target. If it became apparent the hydraulic conditions at the target cross section would not permit a calibration 

within the tolerance limit, an attempt was made to minimize 

the difference then the process was moved on to the next 

upstream cross section.

For the final calibrated model (a total of 2,680 cross 

sections), 88% fell within the target error tolerance of 30 mm. 

Most of the cross sections with error greater than 30 mm were 

confined to rapids or other high-gradient sections of the river. 

These high-gradient sections were problematic for the calibration process. An underlying assumption of the model is that 

flow in the river is one dimensional, but flow in a fan-eddy 

complex is three-dimensional with large vertical velocities in 

the rapid, and two-dimensional eddy structures both upstream 

and downstream from the rapid. Therefore, the target tolerance 

in and immediately downstream from rapids was, at times, 

discarded in favor of accuracy in the pools upstream of the 

rapids. In fact, the standard approach of the calibration process 

near rapids was to first correct the predicted water surface in 

the pool at the head of the rapid, then attempt to match the 

water-surface profile in the core of the rapid as closely as possible. Examining the model calibration only in the pools (that 

is, cross sections where the water-surface slope from the next 

upstream cross section is less that 0.001), 98.0% of the calibrated predictions were within the 30 mm tolerance and 99.8% 

were within 60 mm of the known water-surface elevation. 

Expansion and contraction near rapids, combined with 

the model’s attempt to preserve conservation of mass, resulted 

in troughs (that is, lowered water-surface elevations at the 

constrictions) in the longitudinal water-surface profile. Within 

flumes experiencing controlled and one-dimensional flow 

conditions, these troughs are readily observed at constrictions (Chow, 1959), but troughs have not been observed in 

the natural channel and three-dimensional flow conditions 

of the Colorado River in Grand Canyon. Energy consumption by turbulence, breaking waves, and recirculation eddies 

in rapids create conditions where troughs do not occur. The 

one-dimensional numerical model, not able to replicate the 

complex mechanics of the rapids, tends to predict the presence of troughs, particularly for higher discharges. Users of 

the model need to be aware of these water-surface elevation 

troughs for two reasons: (1) one-dimensional hydraulics do not 

apply to rapids—the model should not be used to make precise 

predictions about the water-surface elevation within rapids—

and (2) the user should post-process all water-surface profiles 

taken directly from HEC-RAS by backfilling all water-surface 

trough from downstream to upstream. More importantly, users 

need to understand that the model is best at predicting the 

water-surface elevation in the pools above rapids, not directly 

in rapids nor in the eddy-dominated flow of the expansion 

below rapids.

Evaluating Synthetic Bathymetry

When typically building standard-step hydraulic models, 

the topography of the river channel, either known or measured, is first built into the model geometry. Next, a reasonable 

approximation of the roughness coefficient is made to enable 

the model to make hydraulic predictions. Finally, if available, 

high-water marks of known discharge are compared to the 

model results, thus suggesting how roughness coefficient may 

be adjusted to improve the model performance. 

In the case of modeling the Colorado River in Grand 

Canyon, where bathymetric data are largely unavailable, we 

were forced first to assume a best guess roughness coefficient, 

then create synthetic bathymetry in order to build the model. 

As far as we know, this approach was used just twice previously, by Randle and Pemberton (1987) and by the Bureau of 

Reclamation (2001), but if this approach of using synthetic 

bathymetry can produced useful results, this technique offers 

a valuable, alternative way to construct hydraulic models 

without knowledge of the bathymetry of the river channel. The 

technique could be particularly useful in modeling bedrockcontrolled rivers where bathymetry is difficult or impossible to 

measure. Because the technique of using synthetic bathymetry 

is both unorthodox and potentially valuable, additional analysis of the quality of the final synthetic bathymetric product 

was warranted. 

Cross sections of the final calibrated model were compared to real bathymetric data measured by Northern Arizona 

University (NAU) in May 2002 through the GCMRC research 

program (Matt Kaplinski, Northern Arizona University, written communication, 2008). These actual bathymetric data were 

not available when the hydraulic model was built and therefore were not incorporated into the model; future versions of 

the hydraulic model could incorporate these data. In all, direct 

comparisons of the synthetic and actual bathymetry were 

made at 169 NAU cross sections located at monitoring reaches 
Estimating Model Accuracy 13

distributed throughout the river corridor. These cross sections 

were measured in pools and slow water between rapids and no 

cross sections were measured directly in rapids. The hydraulic 

radius of the wetted cross section was used to compare the 

data sets. Hydraulic radius is given by (Chow, 1959)

R A

P h = , (8)

where P is the length of the wetted perimeter along the bottom 

of the cross section.

The accuracy of the synthetic bathymetry was also evaluated by comparing predicted water velocity from the hydraulic 

model with water velocity data collected in the 1990s. The 

mean velocity of any cross section in the hydraulic model is 

calculated using equation (2). Combining the mean velocity 

of adjacent cross sections, the predicted mean velocity of the 

model through any reach can be calculated. These predicted 

mean velocity values were then compared to velocity data 

collect at steady low discharge (Graf, 1995) and steady high 

discharge (Konieczki and others, 1997). 

Finally, we conducted a sensitivity analysis on the synthetic bathymetry using variable Manning’s roughness coefficients. Because the selection of roughness coefficient may be 

the largest source of error in generating synthetic bathymetry, 

the intent of the sensitivity analysis was to estimate what 

may have been a better choice of Manning’s n for the river 

channel when constructing the overall synthetic bathymetry. 

Two short reaches of the river were chosen for the sensitivity analysis: the first reach was near Anasazi Bridge between 

river miles 42.669-45.453, a section of the river characterized 

by numerous debris fans and recirculation eddies, and the 

second reach was in Inner Granite Gorge between river miles 

86.567-87.946, a relatively straight, narrow section of the river 

with relatively few recirculation eddies. Using the calibration 

technique described above, three different sets of synthetic 

bathymetry was generated in each reach using Manning’s 

roughness values of n=0.020, 0.035, and 0.050. The synthetic 

bathymetry from each unique calibration was then compared 

to the known NAU bathymetric data described above. In addition to comparing hydraulic radius, the wetted area, A, of the 

cross sections was compared as well as the hydraulic depth, 

given as (Chow, 1959),

D A

T h = , (9)

where T is the top width of the water in the cross section. 

While the labor-intensive nature of constructing the hydraulic 

model precluded recalibration using roughness values suggested by the sensitivity analysis, the analysis results indicate 

how a future modeling effort might create more accurate 

synthetic bathymetry.

Estimating Model Accuracy

The accuracy of the hydraulic model’s prediction is 

affected by prediction errors inherent in the model as well as 

errors built into the DEM used to construct the geometry in the 

hydraulic model. Because the model was calibrated to a flow 

of 227 m3/s, the accuracy of the model at this discharge is presumably as good as the DEM used to build the model plus the 

30 mm calibration tolerance. As discussed earlier, the DEM 

has an absolute vertical error estimated to be less than 0.3 m.

To estimate the accuracy of stage predicted by the final, 

calibrated hydraulic model, a comparison of the model’s output to known stage data measured along the Colorado River in 

Grand Canyon is needed. To estimate error for this study, the 

results from the hydraulic model for discharges up to 2,500 

m3/s were compared to stage-discharge relations measured at 

several monitoring sites by NAU and by Konieczki and others 

(1997); for flows above 2,500 m3/s, model predictions were 

compared to stage-discharge curves from permanent USGS 

gaging stations and to driftwood left at select sites along the 

river.

The NAU stage-discharge data were collected in the 

early 2000s, and the data are reported in NGDV88 ellipsoid 

vertical coordinates, allowing a direct comparison with the 

model’s results. Under a long-term monitoring program, NAU 

routinely measures topography and stage at 47 locations in 

Grand Canyon, constructing stage-discharge relations for the 

sites at flows from 142 to 1,300 m3/s spanning the range of 

flows seen in Grand Canyon over the past 20 years (Hazel and 

others, 2007). Of the 47 locations, 45 were within the domain 

of the hydraulic model and used for error analysis. Using 

high-water marks from the 1983 flood and driftwood left by 

historic floods, NAU was also able to extend the rating curves 

up to 2,500 m3/s at four monitoring sites (Vaseys Paradise, 

river mile 32.17; Palisades, river mile 66.1; Comanche, river 

mile 68.3; 220R, river mile 220.065). These sites were used to 

calculate error from 1,400-2,500 m3/s. For a given NAU site, 

the predicted water-surface elevation from the hydraulic model 

was calculated by linearly interpolating values from the two 

cross sections in the model bounding the NAU site.

The residual, or the difference between the model prediction and the known NAU stage, for a given discharge was 

calculated at each site. But the residual at one location did 

not fully represent the uncertainty of the model’s predictions 

at other sites. To estimate the overall accuracy of the model 

as applied to the entire river, residuals were calculated for 

all NAU monitoring sites and averaged, then the standard 

deviation of the residuals was calculated to get an estimate of 

the confidence intervals of the model prediction for a given 

discharge. The total estimated error for the model was determined to be the standard deviation of the residuals added to 

the residual mean. The model error (residual mean plus the 

residual standard deviation) represented the total error of the 

model prediction to a stated confidence interval. For example, 

if the total error was 1.0 m, then the model would be capable 
14 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

of predicting stage at any location to ±1.0 m with a confidence 

of 68% (that is, one standard deviation). The percent error was 

calculated by normalizing the total error against an estimate 

of relative stage. This estimate of relative stage was calculated 

to be the difference in height between the stage of interest and 

the water-surface elevation at zero discharge (0 m3/s). The 

water-surface elevation at zero discharge was estimated by 

extrapolating the known NAU water-surface elevations at 227 

m3/s and 141 m3/s down to 0 m3/s. 

As a secondary check of the error estimates generated 

using the NAU data, model predictions were also compared 

to stage data collected at 27 sites during the 1996 high-flow 

experiment (Konieczki and others, 1997). These sites were 

distributed widely throughout Grand Canyon (fig. 1) and 

measured stage at a steady flood discharge of 1,322 m3/s and 

a steady post-flood discharge of 237 m3/s. The differences 

in measured stage for these two discharges were then compared to the hydraulic model predictions to calculate residual 

mean and the standard deviation of the residuals based on the 

Konieczki and others (1997) data. 

While other stage data were measured at temporary and 

permanent gaging stations throughout the river corridors 

during the 1980s and 1990s (see Garrett and others, 1993; 

Gauger, 1996; Rote and others, 1997), these data were not 

generally tied to discharge data. In addition, potential changes 

in the water-surface profile in intervening years may increase 

uncertainty in comparing these older stage-discharge data to 

the river profile in 2000-2002. Ultimately, these additional 

data were not of sufficient quality (nor did they offer unique 

information) to merit inclusion in the error analysis.

To estimate error for flows above 2,500 m3/s, stagedischarge curves from USGS gaging stations at Lees Ferry 

(Colorado River at Lees Ferry, Arizona, 09380000) and 

Grand Canyon (Colorado River near Grand Canyon, Arizona, 

09402500), as reported by Topping and others (2003), were 

used. To directly compare these curves, the USGS stage data 

were adjusted vertically to match the DEM water-surface elevation at 227 m3

/s. While data from only two gaging stations 

do not allow a full statistical analysis of the model’s accuracy 

for extreme floods, these stage-discharge data allowed a rough 

estimate of how reasonably the model predicts stage for flows 

up to 5,900 m3

/s. In addition, the model output was analyzed 

against driftwood strand lines and driftwood piles located 

below Palisades Creek and at Granite Park (river mile 209) to 

compare model performance during large flows.

Virtual Shoreline Construction

The first step in developing virtual shorelines of water 

inundation was to produce a dense linear network of stageelevation locations along the centerline of the Colorado River 

from Glen Canyon Dam to Lake Mead. This linear array of 

points was constructed by sub-dividing GCMRC’s 100-meter 

Colorado River centerline coverage (GCMRC, 2002) into 

nearly 73,000 5-meter increments, or centerline points located 

between Lees Ferry and Diamond Creek. Water-profile data 

from the hydraulic model were then projected onto these 

centerline points by linearly interpolating values between the 

bounding upstream and downstream cross sections. Owing to 

the density of cross sections in the model, it was assumed that 

the water-surface elevation between cross sections was linear. 

The second step was to project these water-surface data 

across the width of the river corridor, orthogonal to the centerline points. We constructed a two-dimensional grid of 5-meter 

cells extending across the width of the Canyon and coded each 

cell to its nearest 5-meter centerline point. The water-surface 

data along the centerline were then projected to the appropriate 5-meter cells, enabling the two-dimensional mapping of 

the shoreline at various discharges.

Virtual shorelines and inundation regions were then constructed by subtracting the two-dimensional grid representing 

the water surface from the DEM of the Canyon. The 5-meter 

stage-elevation grid was first re-sampled to 1-meter to match 

the spatial resolution of the DEM. Shorelines were generated 

by converting all negative grid cells into contiguous-area polygons. This three-step methodology was automated as an ARC 

Macro Language (AML) program.

Note that while known vegetation artifacts were removed 

from cross sections used to build the hydraulic model, vegetation stands were not removed from the base DEM used to display the virtual shorelines. As a result, polygons around stands 

of tree can be seen in the resulting inundation maps.

Results

First, the synthetic bathymetry was analyzed and evaluated against (where available) known hydraulic data measured 

in the river. Secondly, the Grand Canyon hydraulic model 

and the accompanying virtual shorelines were tested under a 

number of discharge values ranging from 142 m3/s up to 5,900 

m3/s. For the hydraulic model runs, inundation maps were generated thus showing the location of shorelines along the river 

corridor. A summary of some of the important flow values 

modeled in the study is shown in table 1.

Analysis of Synthetic Bathymetry

Table 2 reports results of the comparison of hydraulic 

radius between known bathymetric data measured by NAU 

and the synthetic bathymetry of the hydraulic model. The 

NAU data were grouped into nine separate reaches, and comparisons made in each of the reaches showed that the hydraulic model consistently predicted a hydraulic radius less than 

reality. Overall, the mean hydraulic radius reported by NAU 

was 4.69 m and the mean hydraulic radius predicted by the 

hydraulic model for the same 169 cross sections was 3.09 m, a 

difference of -1.60 m. In other words, the synthetic bathymetry 

predicted a mean hydraulic radius 34% smaller than reality. 
Results 15

These results indicate that the Manning’s n used to calibrate 

the synthetic bathymetry (n=0.035) was perhaps too small. If 

the roughness coefficient in the real channel is larger than the 

value assumed when calibrating synthetic bathymetry, then 

more energy would need to be consumed in the model such 

that the predicted water-surface profile would closely match 

the actual water-surface profile at 227 m3/s. Therefore, the 

depth in the cross sections of the final calibrated synthetic 

bathymetry would be smaller, resulting in smaller hydraulic 

radii, to compensate for the energy losses needed to predict the 

water surface.

The velocity data show that the predicted velocity values 

of the hydraulic model were 59% greater than the measured 

velocity data for low discharges of about 430 m3/s (table 3). 

Similarly, the predicted velocity values of the model were 

40% greater than actual velocity data for intermediate flows of 

about 1,300 m3/s (table 3). Again, these data indicate that the 

cross sections in the synthetic bathymetry are shallower than 

the cross sections in the actual river. This result also indicates 

that a larger Manning’s n would probably have led to the creation of a more accurate synthetic bathymetry. 

The results of the sensitivity analysis at Anasazi Bridge 

(table 4) further support the conclusions drawn from comparisons of synthetic bathymetry to real bathymetry and the 

velocity data. As Manning’s n increased from 0.020 to 0.050, 

the recalibration of the model showed that in this reach of 

river, the mean water area increased from 194 to 330 m2

, 

the hydraulic depth increased from 1.98 to 3.37 m, and the 

hydraulic radius increased from 1.96 to 3.27 m. In contrast, the 

NAU data show that the actual water area is 521 m2

, the actual 

hydraulic depth is 5.13 m, and the actual hydraulic radius is 

4.89 m. More telling, when the results of the sensitivity analysis are extrapolated to larger roughness values, a Manning’s 

n of 0.092 would be needed by the model to match the actual 

water area, a Manning’s n of 0.087 would be needed to match 

the actual hydraulic depth, and a Manning’s n of 0.086 would 

be needed to match the actual hydraulic radius. As a function 

of river mile, figure 8 shows the hydraulic radius for each 

sensitivity calibration as well as the actual hydraulic radius 

measured by NAU. Except for the cross sections upstream of 

river mile 43.4, the results of the sensitivity analysis show the 

synthetic bathymetry is typically smaller than the actual river. 

Note that some cross sections were located in rapids and the 

bathymetry here was not measured by NAU.

The results of the sensitivity analysis for the reach in 

Inner Granite Gorge are shown in table 5. In contrast to the 

river near Anasazi Bridge, the river in Inner Granite Gorge is 

steep and narrow, with few recirculation eddies. But similar to 

the results of the sensitivity analysis for the river near Anasazi 

Bridge, the sensitivity analysis in Granite Gorges shows that 

a Manning’s n significantly larger than 0.050 would probably 

be needed to create a synthetic bathymetry that accurately 

simulates the hydraulic parameters of the river. For example, 

by extrapolating roughness values, it appears a Manning’s n

of 0.075 would be needed to match the measured water area 

of 377 m2

, a Manning’s n of 0.074 would be needed to match 

the measured hydraulic depth of 6.48 m, and a Manning’s n

of 0.068 would be needed to match the measured hydraulic 

radius of 5.38 m.

The evaluation of the synthetic bathymetry using known 

bathymetry, velocity data, and the two sensitivity analyses 

suggest that if the model were going to be constructed again, 

a larger roughness coefficient could enable the synthetic 

bathymetry to more accurately match the bathymetry in the 

actual river. The need for a larger roughness coefficient in the 

model for calibrating synthetic bathymetry does not result 

from roughness elements in the channel requiring a larger 

Manning’s n—based on the bed material and morphology, 

the choice of Manning’s n of 0.035 is reasonable for the main 

channel. Instead, there are other sources of energy losses in the 

hydraulic system requiring the model to have a larger overall 

roughness coefficient during calibration. For example, the 

ubiquitous nature of recirculation eddies in the actual river 

may require the synthetic bathymetry to be shallower than 

reality in order to accommodate the observed energy losses. 

Also possible, inaccuracies in the water-surface profile of the 

river used to calibrate the synthetic bathymetry may result in 

a bias built into the synthetic bathymetry. If the water-surface 

profile used for calibration, for example, was biased in such 

a way that the pools were steeper than reality, the synthetic 

bathymetry would need to be made shallow to compensate 

for energy losses in the pools suggested by the steeper profile. 

Regardless of the reason for the inaccuracy in the synthetic 

bathymetry, if a higher roughness coefficient were used to 

Table 1. Significant flows simulated in this study.

Discharge (m3/s) Discharge (cfs) Significance

227 8,000 Typical flow during 

GCMRC remotesensing overflights

566 20,000 Typical high flow 

in dam-regulated 

regime

1,270 45,000 Approximate size 

of the 1996 Controlled Flood

2,750 97,000 Approximate size of 

the 1983 flood, 

the largest postdam flood

3,000 106,000 Approximate size of 

the 1958 flood

3,500 125,000 Approximate size of 

the 1957 flood, 

the most recent 

pre-dam flood

4,800 170,000 Estimated size of the 

1921 flood1

5,900 210,000 Estimated size of the 

1884 flood1

1

As reported by Topping and others (2003)
16 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

Table 2. Mean hydraulic radius in nine reaches as measured by Northern Arizona University and as predicted by the Grand Canyon 

hydraulic model.

River Miles Number of cross 

sections

Mean hydraulic radius, measured (m)Mean hydraulic 

radius, predicted (m) Difference (m) Percent difference

1.105-2.638 16 3.89 3.12 -0.77 -20%

21.888-23.664 23 4.76 2.60 -2.16 -45%

29.881-32.000 28 5.19 2.96 -2.23 -43%

42.669-45.453 20 4.89 3.42 -1.47 -30%

54.484-56.271 13 3.44 2.94 -0.50 -14%

63.569-66.083 20 4.75 2.99 -1.76 -37%

86.567-88.097 13 5.16 3.72 -1.44 -28%

119.319-122.397 26 4.90 3.29 -1.61 -33%

208.038-209.101 10 4.34 2.86 -1.48 -34%

Total 169 4.69 3.09 -1.60 -34%

Table 3. Comparison of measured and model-predicted water velocity at low flow and high flow for the Colorado River in Grand 

Canyon.

River Miles Q (m3/s) Measured Velocity 1 (m/s) Predicted velocity (m/s) Difference

0.0-35.9 425 0.87 1.44

35.9-61.1 425 0.75 1.28

61.1-76.6 430 1.1 1.43

76.6-117.6 433 0.97 1.83

117.6-166.5 436 1.1 1.71

166.6-213.5 436 1.1 1.64

Mean 0.98 1.56 +59%

0.0-7.9 1276 1.51 1.91

7.9-35.9 1276 1.69 2.59

35.9-61.1 1276 1.49 2.15

61.1-76.6 1327 1.9 2.40

76.6-122.0 1327 1.93 3.07

122.0-166.5 1327 1.89 2.89

166.5-183.0 1327 2.11 2.51

183.0-225.0 1327 1.83 2.61

Mean 1.79 2.52 +40%

1

As reported by Graf (1995) for low flow and Konieczki and others (1997) for high flow.
Results 17

(YDRAULIC2ADIUSM	

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Figure 8. (Top) The water-surface profile near Anasazi Bridge showing President Harding Rapids and several downstream rapids. 

(Bottom) Results of the sensitivity analysis near Anasazi Bride showing the actual hydraulic radius measured by NAU and the hydraulic 

radius as predicted by the model calibrated for Manning’s n values of 0.020, 0.035, and 0.050. The vertical gray bars shows areas where 

no NAU data are available.

Table 4. Change in mean hydraulic parameters for a discharge of 227 m3/s as a function of Manning’s roughness coefficient for the 

reach near Anasazi Bridge (river miles 42.669 to 45.453). Also shown are the hydraulic parameters measured from available bathymetric 

data and the approximate roughness needed by the hydraulic model to match the measured data. 

Manning’s Roughness Water Area 

(m2

)

Hydraulic Depth 

(m)

Hydraulic Radius 

(m)

0.020 194 1.98 1.96

0.035 264 2.73 2.67

0.050 330 3.37 3.27

Measured data: 521 5.13 4.89

Roughness needed to match 

measured data:

0.092 0.087 0.086
18 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

Table 5. Change in mean hydraulic parameters for a discharge of 227 m3/s as a function of Manning’s roughness coefficient for the 

reach of river in Inner Granite Gorge (river miles 86.567 to 87.946). Also shown are the hydraulic parameters measured from available 

bathymetric data and the approximate roughness needed by the hydraulic model to match the measured data. 

Manning’s Roughness Water Area 

(m2

)

Hydraulic Depth 

(m)

Hydraulic Radius 

(m)

0.020 169 2.97 2.84

0.035 237 4.17 3.85

0.050 282 4.93 4.44

Measured data: 377 6.48 5.38

Roughness needed to match 

measured data:

0.075 0.074 0.068

Figure 9. Virtual shorelines at Eminence Break Camp (river mile 44.5). Vegetation surface elevations can be clearly seen as island 

polygons.

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Results 19

recalibrate the synthetic bathymetry, the performance of the 

model for flows larger than 227 m3/s suggests lower Manning’s n values (on the order of 0.025 to 0.035) would be 

needed with rising discharge in the main channel and the 

overbanks. Such variable roughness values, changing spatially 

and with discharge, would greatly increasing the complexity 

of the model, and would probably be impractical for a model 

of this size.

Nonetheless, the reader needs to be aware that the 

synthetic bathymetry is 30-40% shallower than the actual 

river and that the predicted mean velocity in the reaches are 

as much as 60% faster than the velocity found in the river. 

Despite the inaccuracies in the synthetic bathymetry, the 

model can still predict stage to some level of accuracy, the 

quantification of which will be discussed below.

Inundation Maps

Figure 9 shows an example of the virtual shorelines 

generated from the hydraulic model at Eminence Break Camp 

near river mile 44.5. The flow in the river is from top to bottom and a relatively flat deposition bar on river left shows a 

large change in shoreline position with a modest rise in stage. 

The DEM contains a combination of bare ground and vegetated surface elevations, which are not true ground elevations 

but instead are artifacts of DEM construction. Virtual shoreline 

intersections with vegetated areas were readily apparent, as 

they formed island polygons and other convex patterns, visible 

on the map for Eminence Break Camp (fig. 9). For this site, 

the hydraulic model tended to under predict the water-surface 

elevation for flow greater than 400 m3/s. The largest residual 

was -0.32 m for a discharge of 878 m3/s, a percentage error 

of 10%. At a discharge of 1,274 m3/s, the residual, is -0.25 m 

(fig. 10). 

Comparison of the model predictions with other NAU 

sites gave better results. At Dino Camp, for example, the 

model showed good agreement with the NAU data (fig. 11). 

Similar to the results at Eminence Break Camp (fig. 9), the 

model under predicted the water surface for intermediate flows 

of 500-1,200 m3

/s. The largest residual was -0.16 m at 878 

m3/s, representing an error of 4%. At 1,274 m3/s, the model 

under predicted the water surface by -0.08 m (5% error). The 

virtual shorelines at Dino Camp, therefore, represent a shoreline map that is relatively close to actual shorelines at most 

discharges (fig. 12).

Accuracy Estimate for flows under 1,400 m3/s

Figure 13 shows the residual mean and standard deviation of the residuals of the hydraulic model for a range of 

discharges calculated from the NAU sites; table 6 lists the 

data used to construct the figure. Figure 13 shows that the 

mean of the residuals is smallest for a discharge of 283 m3/s. 

At the calibration discharge of 227 m3/s, the residual mean 

was 0.03 m, the standard deviation of the residuals is 0.30 m, 

and with 68% confidence, the hydraulic model predicts stage 

in the river to within ±0.32 m. The model accuracy was ±0.64 

m at 878 m3

/s, when flow begins to inundate the overbank 

regions. Heavy riparian vegetation on the lower sections of the 

overbanks results in a larger roughness and a higher watersurface elevation than the model predicts. The model, using 

a constant 0.035 roughness coefficient, may fail to capture 

energy losses from extra vegetation on the overbanks. Another 

potential source of error is recirculation eddies along the river 

corridor. The presence of eddies, particularly upstream and 

downstream from debris fans, act to both consume energy in 

.!5SITERATINGCURVE

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Figure 10. Comparison of the hydraulic model predictions and 

the stage-discharge data as measured by NAU at Eminence Break 

Camp near river mile 44.5.

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Figure 11. Comparison of the hydraulic model predictions and 

the stage-discharge data as measured by NAU at Dino Camp near 

river mile 50.
20 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

the flow and constrict the conveyance of the channel. Because 

the hydraulic model is one-dimensional, it cannot account for 

the recirculation eddies, potentially affecting model accuracy. 

Also, as the discharge in the Colorado River in Grand Canyon 

increases from 300 m3/s up to larger intermediate discharges 

near 1,000 m3/s, many rapids wash out (become less steep and 

experience a reduction in the total elevation drop through the 

rapid) as downstream features exert hydraulic control over 

these rapids. In the river, more energy is consumed in strong 

turbulence within the pools, and less energy is consumed by 

the constriction of rapids. The hydraulic model, which under 

subcritical flow regime consumes large amounts of energy in 

the rapids at low discharge, may potentially under predict the 

energy consumption in pools, resulting in an under prediction 

of the stage. 

A comparison of the results of the hydraulic model to 

the stage data reported by Konieczki and others (1997) shows 

that for a change in discharge from 237 m3/s to 1,322 m3/s, the 

mean of the residuals of the 27 sites indicate the model under 

predicted stage by -0.14 m. The standard deviation of these 

residuals is 0.39 m indicating an overall accuracy of ±0.53 

m. These error estimates using the data from Konieczki and 

others is consistent with the accuracy estimates using the NAU 

data.

To evaluate the sensitivity of the model to different 

values of Manning’s n, global roughness values of 0.025 to 

0.045 were evaluated and compared to the standard n=0.035 

(fig. 14). At the smallest discharge evaluated, 142 m3/s, average error changed roughly 0.30 m as roughness is raised from 

n=0.025 to n=0.045. At the largest discharge, 1,274 m3/s, average error changed roughly 0.88 m over the span of roughness 

values evaluated. For flow between 700 and 1,300 m3/s, figure 

14 indicates a global roughness of n=0.040 slightly improves 

model performance, suggesting the model underestimated 

energy loss at moderate discharges from either overbank vegetation or larger eddies in the flow. 

Modeling Floods between 1,400 and 2,500 m3/s

The residuals as a function of discharge up to 2,500 m3/s 

for the four NAU sites and two USGS gaging stations are 

shown in figure 15. At 1,700 m3/s, the residuals and standard 

deviation from the six sites is 0.12±0.67 m. At 2,500 m3/s, the 

mean of the residuals is 0.20±0.76 m giving an overall accuracy of about 1.0 m. 

The model agreed well with the two USGS gaging stations with the residuals under 0.5 m. The model also agreed 

well with the NAU data reported at river mile 220. This 

NAU rating curve for this site was constructed using highwater marks from the 1983 flood (2,750 m3

/s). Similarly, the 

rating curve at Vaseys Paradise was constructed with 1983 

high-water marks, but the hydraulic model under predicted 

stage by about one meter. This section of river is in a narrow 

reach of canyon and the model estimate of n=0.035 may not 

represent the true energy losses. In contrast, the model tends 

$INO#AMP .

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Figure 12. Virtual shorelines at River Mile 50. The effects of the 

vegetated surface in the DEM can be seen as island polygons.

Table 6. Error measured relative to all NAU monitoring sites 

at varying discharge with a constant roughness coefficient of 

n=0.035.

Discharge 

(m3/s)

Residual 

mean 

(m)

Standard 

deviation 

of the 

residuals 

(m)

Total estimated error 

(m)

Percent 

error

142 -0.03 0.34 0.37 45%

227 0.03 0.30 0.32 25%

283 0.02 0.29 0.30 19%

425 -0.06 0.29 0.35 15%

566 -0.15 0.31 0.46 15%

708 -0.23 0.33 0.56 16%

850 -0.29 0.34 0.63 15%

878 -0.30 0.35 0.64 15%

991 -0.31 0.36 0.67 15%

878 -0.29 0.37 0.66 13%

1,274 -0.24 0.38 0.62 12%
Results 21

Figure 13. The average of the residuals of the hydraulic model predictions compared against stage-discharge data measured at the 

NAU monitoring sites. The error bars represent the standard deviation of the residuals.













0REDICTION%RRORM	

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Figure 14. Sensitivity analysis of the model residual mean (averaged for all NAU sites) for different values of Manning’s n.

Figure 15. The residual error calculated at six sites for discharges from 1,400 to 2,500 m3

/s.
22 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

Figure 16. Matched photos showing the section of river at Boulder Narrows (river mile 18.746) during (top) 3,570 m3/s flood on 

6/11/1957 with P.T. Reilly rowing (Duane Norton, NAU.PH.97.46.115.63, courtesy of Northern Arizona University, Cline Library, Special 

Collections and Archives, P.T. Reilly Collection) and (bottom) low flow of roughly 340 m3

/s photographed 3/1/2005 (Steve Young, Desert 

Laboratory Repeat Photography Collection, stake 4810).
Results 23

to over predict stage at the Palisades and Comanche sites. 

The NAU rating curves above 1,300 m3/s for these sites were 

constructed from driftwood data (Draut and others, 2005). 

It may be that within this geomorphically wide reach of the 

river, the roughness coefficient needed to model the river 

is lower than 0.035. The river in Furnace Flats is relatively 

wide with a lesser gradient than other sections of the river. As 

the water depth increases with rising stage, roughness elements on the bed become smaller in a relative sense and the 

overall roughness coefficient, in theory, drops. For example, 

in Glen Canyon, the Bureau of Reclamation (1957) reported 

Manning’s n decreased from 0.03 to 0.02 as flow increased 

from 283 to 2832 m3/s. The other possible explanation for the 

apparent over prediction of stage is that debris flow activity 

in this reach of the river has significantly aggraded the river 

bed. For example, at least four debris flows occurred at either 

Lava Canyon or Palisades Creek (river mile 66.0) between 

1965-1984 (Cooley and others, 1977; Melis and others, 1994; 

Webb and others, 2000), two debris flows occurred at Comanche Creek (river mile 67.7) between 1999-2001, creating a 

rapid, and one large debris flow occurred at Tanner Creek in 

1993 (Melis and others, 1994). These debris flows and associated boulders raised the river. At Tanner Rapid, Melis and 

others (1994) report a rise of 1.0 m in 1993 that was lowered 

0.27 m by a flood in 1996 (Webb and others, 1999). Though 

the rise in the water-surface elevation at Comanche and at 

Palisades was not measured, the size and number of debris 

flows and the observation of a new rapid at Comanche suggest 

the river also rose in this reach by an amount similar to Tanner 

Rapids. Because the hydraulic model was built with modern 

topography, aggraded with fresh debris-flow material, the predicted water surface is probably higher than the water surface 

20 years earlier, possibly explaining the over prediction with 

respect to driftwood strand lines.

Modeling Large Floods

Photographic evidence of historic floods can be used to 

verify the accuracy of the hydraulic model for large discharges. For example, a photograph taken during the 3,570 

m3

/s flood in 1957 shows the water level and P.T. Reilly navigating his boat through Boulder Narrows (river mile 18.746) 

in Marble Canyon (fig. 16). This 1957 flood was the largest in 

Grand Canyon in the past 79 years. Boulder Narrows is a wellknown section of the Colorado River where a large boulder on 

the order of 10-20 m fell into the center of the river channel; 

at low discharge, the river flows to either side of the boulder 

whereas the boulder is overtopped during large discharge (fig. 

16). When this section of the river was simulated with the 

hydraulic model, the predicted water-surface elevation fell 

roughly one meter below the top of the boulder (fig. 17). The 

predicted water surface of the river for a flood of 3,570 m3

/s 

is just below the top of the boulder at Boulder Narrows, but 

because the model is one-dimensional, it does not account for 

localized backwater effects of the flowing water as it backs up 

behind the boulder. If the localized backwater effect was considered (for example, in a two-dimensional model) the model 

would better predict the incipient overtopping of the boulder 

shown in figure 16. In fact, the elevation of the stage plus 

the velocity head at this cross section in the model is 0.48 m 

above the top of the boulder, confirming that with impinging 

flow, the model does predict the over topping of the boulder.

Driftwood lines from large pre-dam floods are visible 

today throughout Grand Canyon. Based on the driftwood lines 

left by large historic floods near Palisades Creek at river mile 

66 (David Topping, USGS, written communication, 2005), the 

hydraulic model over predicted the water-surface elevation in 

this reach by 1.11 m at 5,000 m3

/s and 1.40 m at 5,900 m3/s 

(fig. 18). This over prediction by the model may be related 

to the choice of a Manning’s roughness coefficient that may 

be too large for this section of the river. The hydraulic conditions of the river, however, have also changed between 1921 

and 2000. As mentioned above, high debris-flow activity in 

this section of the river during the post-dam era has aggraded 

the river channel. Also important, a significant increase in the 

quantity of riparian vegetation below the 3,000 m3

/s water 

line (Webb, 1996; Webb and others, 2007) has changed the 

roughness elements along the overbanks. With these changes 

both in vegetation and the topography on debris fan, it is probable that if a flood of 4,800 m3

/s or 5,900 m3/s came down the 

river today, the water surface would be higher than high-water 

marks indicated by the driftwood strand lines left in 1921 and 

1884, though that rise might not be as great as 1.11 m or 

1.40 m.

More recently, numerous driftwood strand lines and 

driftwood piles were left throughout the river corridor from 

the 3,570 m3

/s flood in 1957. Granite Park, located near river 

mile 209, is a relatively wide expanse of river corridor caused 

by faulting of the Granite Park Fault (fig. 19). Just upstream of 

Figure 17. Hydraulic model cross section at river mile 18.746 

(Boulder Narrows) showing predicted water-surface elevation at 

227 m3

/s (the shaded area) and the predicted high water mark of 

the 3,570 m3

/s flood that occurred in Grand Canyon in 1957. The 

peak of the actual flood overtopped the boulder (fig. 14); the model 

under predicts the stage of the flood high-water mark by 

about 1 m.

920

870

880

910

900

890

930

Elevation (m)

Cross Section Station (m)

0 20 40 60 80 100 120

High water mark: 

3,570 m3/s
24 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

0ALISADES

#REEK

.

MS

M

S

M

S

M

S

M

S

M

S

METERS

  

M

S

M

S

M

S

M

S

M

S &LOW

Figure 18. Virtual shorelines (shown as lines) and driftwood strands (shown as dots) for different floods at Palisades Creek (river mile 

66) generated with a Manning’s n value of 0.035. The results indicate that for this section of river, the model over predicts the watersurface elevation for flows above 4,800 m3

/s by as much as 1.4 m.
Results 25

209.584 

209.542 

209.501 

209.445 

209.368 

209.327 

209.288 

209.229 

209.194 

209.185 

209.164 

Flow 

1958 drift

1957 drift

-ODELPREDICTIONATMS















%LEVATIONM	



#ROSS3ECTION3TATIONM	

         

,OCATIONOFDRIFTWOOD

HIGH
WATERMARKS

7ATERSURFACEATMS

Figure 19. Reach of river in Granite Park (river mile 209) showing the broad boulder bar formed from reworked particles entering from 

the tributary on river left. The flow of the river in the image is from top to bottom. Collections of driftwood left by the 1958 (3,000 m3/s) 

flood and the 1957 (3,570 m3

/s) flood are visible toward the middle of the boulder bar near cross section 209.445; these two driftwood 

piles are labeled.

Figure 20. Hydraulic model prediction of the water-surface elevation at river mile 209.445. The inundation of 227 m3

/s is shown as the 

gray area. The high water mark of the 1957 flood (3,570 m3

/s) is also shown illustrating the prediction accuracy for high discharges for 

this reach of river.
26 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

Granite Park 

N 

227 m3/s 

5,900 m3

/s 

4,800 m3

/s 

3,500 m3

/s 

2,750 m3

/s 

1,270 m3/s 

708 m3

/s 

meters 

0 100 200 

Flow 

Figure 21. Virtual shorelines at Granite Park (river mile 209) as predicted by the hydraulic model. The line A-A’ extends between the 

708 and 1,274 m3/s shorelines and rises 1.9 meters across a distance of 100 meters.
Results 27

the expansion, three closely spaced tributaries dump coarsegrained sediment into the river that has been reworked into 

a large boulder-strewn island, which rises in elevation in the 

downstream direction. At low flows, the river flows entirely 

around the right edge of the island. With increasing discharge, 

the river flows around the island with its highest point at the 

farthest downstream point of the bar. Extremely large flows 

(greater than 5,500 m3/s) completely submerge the island. 

The floods of 1957 (3,570 m3/s) and 1958 (3,000 m3/s) did 

not overtop the bar and left large driftwood piles in the lower 

middle section of the island. The dark gray splotches shown 

on figure 19 at cross section 209.445 is the 1958 driftwood 

pile and the larger gray splotch just downstream is the 1957 

driftwood pile (David Topping, USGS, written communication, 2008). In the hydraulic model for 3,750 m3

/s, the cross 

section at river mile 209.368 was completely submerged while 

the cross section at river mile 209.501 still showed a prominent island. The predicted water surface just touched the top 

of the boulder bar at cross section 209.445 (fig. 20), roughly 

predicting the location of the driftwood piles left in 1957.

Mapping the virtual shorelines is useful at Granite Park, 

with the presence of the wide, flat boulder bar. Figure 21 show 

various inundation shorelines at Granite Park from 227 m3/s 

up to 5,900 m3/s. The horizontal accuracy of virtual shoreline 

placement in the inundation maps was affected by three factors, (1) the vertical accuracy of the stage at any given cross 

section, (2) the error associated with bare-ground elevations 

contained within the DEM, and (3) the slope of the ground 

surface across any area of interest. This last factor is of particular concern since relatively small vertical discrepancies can 

result in sizeable line-placement differences across surfaces of 

low slope, as seen at Granite Park between the 708 and 1,274 

m3/s shorelines. Here, there is a 1.9 m vertical stage difference between the two shorelines, but the horizontal distance, 

from north to south, is at least 100 m. Thus, when horizontal 

placement accuracy of the shoreline is an issue, care should be 

taken in evaluating the stage-discharge relation at the particular location.

To more rigorously test the predictive accuracy of the 

hydraulic model for large floods, model results were compared to the stage-discharge relations established at the USGS 

gaging stations at Lees Ferry and Grand Canyon. Comparison of the model results with data from the gaging station at 

Grand Canyon shows how well the model accurately predicts 

stage (fig. 22). Up to 2,500 m3

/s, the prediction error is less 

than -0.20 m. The greatest prediction error -0.34 m occurs 

for flows between 3,000-5,000 m3/s, representing a percentage error of 4%. The error at 5,900 m3

/s is -0.23 m. To test if 

other roughness values may give better results for large floods, 

the model was run with higher Manning’s n (fig. 22). For the 

gage at Grand Canyon, it appears that a Manning’s n value 

of 0.040 would better predict the water-surface elevation for 

floods approaching 5,900 m3/s, with an error of 0.07 m. This 

observation runs counter to speculation that for deeper flow 

depths, the relative size of roughness elements in the channel 

would decrease and the model would require smaller roughness to accurately predict the water surface. The gaging station 

at Grand Canyon is in Upper Granite Gorge. This reach has a 

narrow, confined bedrock corridor. The overall channel slope 

is also relatively steep in this section of river (Hanks and 

Webb, 2006). As stage rises with discharge near this gaging 

station, the river appears to expend energy both from flowing 

against the narrow bedrock walls and from enhanced energy 

consumption in the narrow gorge resulting from turbulent 

eddies and waves breaking on the water surface.

At the Lees Ferry gage, the hydraulic model accuracy 

was good for smaller flows, but the model tended to over 

predict the water-surface elevation for discharge above 2,000 

m3

/s (fig. 23). The model prediction error was generally less 

than 0.10 m at discharges less than 2,000 m3/s. Error steadily 

increased with discharge, reaching a value of 0.69 m at 5,900 

m3/s. When a roughness sensitivity analysis was performed, 

the model accuracy improved if the Manning’s n value was 

decreased to 0.030 for a discharge above 4,000 m3/s, but the 

overall model accuracy for flows between 2,000 and 4,000 

m3/s was about equivalent for roughness values of 0.030 and 

0.035 (fig. 23).

Though independent stage-discharge relations for high 

discharge were only available at these two long-term gaging 

stations, the results of the analysis suggest that model accuracy is generally within 1.0 m for discharges up to 5,900 m3

/s. 

The model appears to better predict stage in narrow, confined 

sections of the river though more analysis is needed to fully 

evaluate the model’s performance in geomorphically wide secFigure 22. The stage-discharge curve for the gaging station for 

Colorado River near Grand Canyon, Arizona (09402500) compared 

to the predicted stage-discharge curves from the hydraulic model 

for n=0.035, n=0.040, and n=0.045 (gage data taken from Topping 

and others, 2003). At a discharge of 5,900 m3

/s, the difference in 

stage between the rating curve and model prediction for n=0.035 

is -0.23 m. For this location, a roughness value of n=0.040 seems to 

be the best value for model prediction.

















      

$ISCHARGEMS	

3TAGEAT'RAND#ANYON'AGEM	

'RAND#ANYON'AGE

'AGINGSTATIONRATINGCURVE

-ODELPREDICTION

N

N

N
28 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

tions of the river (for example, Furnace Flats near river mile 

68).

The hypothesis that smaller Manning’s n value would be 

needed to accurately model large floods may be valid in some 

sections of the river, but the overall set of evidence suggests 

the choice of Manning’s n of 0.035 is largely appropriate for 

most sites even at large discharge. While a slightly lower 

roughness coefficient at Lees Ferry was justified, data from 

the gaging station near Grand Canyon suggest that larger 

Manning’s n values for extreme flows are also justified. For 

geomorphically wide reaches, a lower Manning’s n value is 

justified for limited cases. Moreover, the qualitative data from 

Palisades suggest a low roughness value may be needed in the 

wide geomorphic reach of Furnace Flats (near river mile 68), 

but in western Grand Canyon, data from Granite Park—also 

located in a geomorphically wide reach—indicate that a Manning’s n of 0.035 is appropriate. The specific results at Palisades may be explained by changes in the hydraulic conditions 

along the river corridor or the two-dimensional nature of the 

recirculation eddy below the rapid which cannot be modeled 

accurately by the one-dimensional hydraulic model. Looking at the overall results of the study, a Manning’s n value of 

0.035 is probably the best roughness coefficient overall for the 

Colorado River Grand Canyon.

Suggested roughness values for flows below 

1,400 m3

/s

All flows in the Colorado River in Grand Canyon 

since 1987 have been below 1,400 m3/s. With the quality of 

stage-discharge data collected by NAU, the roughness values 

in the channel and overbanks used when running the final, calibrated model can be adjusted to improve the accuracy of the 

model for these smaller flows. (While the synthetic bathymetry of the model was initially calibrated using a Manning’s n 

of 0.035, the final model can be easily run using any specified roughness value.) Using these NAU data, recommended 

values of Manning’s roughness coefficient were determined 

for selected flows up to 1,274 m3

/s (table 7). Also, figure 24 

shows the mean and the standard deviation of the residuals for 

each of the flows listed in table 7. Using these adjusted roughness values, the accuracy of the model is better than 0.40 m for 

all discharges below 1,400 m3/s.

Future Model Adaptations

The hydraulic model built for Grand Canyon is useful 

and reasonably predicts flow up to 5,900 m3/s. As discussed 

throughout this report, however, there are opportunities to 

improve the model. Cross sections in wide or strongly sinuous 

sections of the river could be regenerated into broken segments to better model flow perpendicular to each cross section, taking care that adjacent cross sections do not intersect. 

The roughness coefficient at each cross section can be better 

Figure 23. The stage-discharge curve for the gaging station 

for Colorado River at Lees Ferry, Arizona (09380000) compared to 

the predicted stage-discharge curve from the hydraulic model 

for n=0.025, n=0.030, n=0.035, and n=0.040 (gage data taken 

from Topping and others, 2003). At a discharge of 5,900 m3

/s, the 

difference in stage between the rating curve and model prediction 

for n=0.035 is 0.69 m. For this location, a roughness value of 

n=0.030 seems to be the best value for model prediction.

'AGINGSTATIONRATINGCURVE

-ODELPREDICTION

N

N

N

N

,EES&ERRY'AGE















      

$ISCHARGEMS	

3TAGEAT,EES&ERRY'AGEM	

Table 7. Error measured relative to all NAU monitoring sites 

for discharge up to 1,274 m3/s with the recommended roughness 

coefficients.

Discharge 

(m3

/s)

Recommended 

Manning’s n

Residual 

Mean 

(m)

Standard 

deviation 

of the 

residuals 

(m)

Percent 

Error

142 0.037 0.00 0.34 42%

227 0.034 0.01 0.29 23%

283 0.034 -0.01 0.28 18%

425 0.037 0.00 0.29 12%

566 0.039 -0.01 0.31 11%

708 0.041 0.00 0.34 9%

850 0.042 0.00 0.36 9%

878 0.042 0.00 0.36 9%

991 0.042 0.00 0.38 8%

1,161 0.041 -0.01 0.38 8%

1,274 0.040 0.00 0.39 7%
Conclusions 29

estimated based on local conditions in the channel and the 

overbanks. The roughness coefficient could also vary as a 

function of discharge and reach type (that is, narrow versus 

wide). Based on the results of the analysis of the synthetic 

bathymetry, a larger Manning’s n value in the channel could 

have enabled the synthetic bathymetry to more closely match 

the real bathymetry. The model’s relative accuracy above 

227 m3/s suggests, however, that if a larger Manning’s n was 

used to create synthetic bathymetry, smaller roughness values 

would be needed for larger discharges and in the overbank 

regions, greatly increasing the complexity of the model. As 

it becomes available, real bathymetry measured from the 

river can be incorporated into the model. If significant debris 

flows impact the river in the future, the synthetic bathymetry 

at the location of the debris flow can be updated to capture 

changes to the water-surface profile. Instead of using universal 

values for expansion and contraction coefficients, the coefficients at each cross section could be tailored to match the 

hydraulic conditions for that location. And finally, HEC-RAS 

has options to insert areas of ineffective flow into the model. 

To better capture the hydraulic behavior below debris-fan or 

bedrock constrictions, ineffective flow regions could be placed 

in the locations of the recirculation eddies. 

Conclusions

A hydraulic model using GIS-based techniques to visualize shoreline inundation was constructed for 364 km of the 

Colorado River in Grand Canyon. The model simulates flood 

events up to 5,900 m3/s using 2,680 cross sections. The model 

was built using topography collected in 2000 and 2002, and 

the model was built in HEC-RAS. The GIS-based techniques 

take the output of the hydraulic model and map inundation 

along the river corridor for any location between Lees Ferry 

and Diamond Creek. 

Based on independently collected stage-discharge data 

from NAU and the USGS, the total error of predicted watersurface elevations from the model is about 0.30 m for flows 

less than 425 m3/s. For flows between 425-1,300 m3/s, error is 

about 0.40 m using the recommended roughness coefficients. 

For flows between 1,300-2,500 m3/s, error is about 1.0 m. For 

floods up to 5,900 m3/s, the model is a reasonable predictor of 

stage with an error of about 1.5 m (roughly 15%). Anecdotal 

analyses of the 1957 flood deposits at Boulder Narrows and 

Granite Park show that the model produces a comparable stage 

and appears to reasonably predict large flows in confined or 

narrow river reaches while possibly over predicting stage in 

wide reaches of the river corridor. 

Finally, the approach of generating synthetic bathymetry 

in the place of actual bathymetry when actual bathymetry is 

difficult to obtain is both unique and unorthodox. Analyses in 

this report show that while the approach can produce a useful model, the careful choice of roughness coefficient before 

calibrating the synthetic bathymetry can improve the value 

of the model by enabling a more accurate guess of the true 

bathymetry.

Figure 24. The average of the residuals of the hydraulic model predictions compared against stage-discharge data measured at the 

NAU monitoring sites when then Manning’s roughness coefficient has been tuned to produce optimal model results. The error bars 

represent the standard deviation of the residuals.













0REDICTION%RRORM	

$ISCHARGEMS	

       
30 Modeling Water-Surface Elevations and Virtual Shorelines for the Colorado River in Grand Canyon, Arizona

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