U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2015–5034

Reducing Cross-Sectional Data Using a Genetic Algorithm 

Method and Effects on Cross-Section Geometry and SteadyFlow Profiles

Genetic

Algorithm

Schematic

Stream cross section

Elevation

Distance

Original

cross

section

GA-reduced

cross section

GA-reduced

chromosome

Selection

crossover

mutation

Initial

population

New

population

Gene

Reducing Cross-Sectional Data Using a 

Genetic Algorithm Method and Effects on 

Cross-Section Geometry and Steady-Flow 

Profiles

By Charles Berenbrock

Scientific Investigations Report 2015–5034

U.S. Department of the Interior

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

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2015

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Suggested citation:

Berenbrock, Charles, 2015, Reducing cross-sectional data using a genetic algorithm method and effects on crosssection geometry and steady-flow profiles: U.S. Geological Survey Scientific Investigations Report 2015–5034, 16 p., 

http://dx.doi.org/10.3133/sir20155034.

ISSN 2329-132X (online)
iii

Contents

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

Introduction.....................................................................................................................................................1

Reduction Methods .......................................................................................................................................2

Standard Reduction..............................................................................................................................3

Genetic Algorithm Reduction..............................................................................................................3

Comparison of Reduction Results...............................................................................................................5

Visual Analysis of Cross-Section Reductions ..................................................................................6

Hydraulic Modeling Analysis of Reduction Methods ...................................................................11

Summary........................................................................................................................................................14

References Cited..........................................................................................................................................15

Figures

1. Graph showing fitness calculation for a hypothetical cross section ..................................4

2. Graphs showing effects of data-point reduction on cross-sectional shape......................7

3. Graphs showing effects of data-point reduction on cross-sectional area ........................8

4. Graphs showing reduction-error (RE) curves for 10 cross sections resulting 

from wwthe genetic algorithm (GA)-reduction method, Kootenai River, Idaho...............10

5. Graphs showing the effects of reduced cross sections on simulated

water-surface elevation at five river discharges..................................................................12

6. Graphs showing comparisons between the original and 10 point, 20 point, and

30 point reduced cross sections produced by two reduction methods for

cross-section 154.575.................................................................................................................13

Tables

Table 1. Comparison of best fitness between several reduction methods for 10 cross 

sections on the Kootenai River, Idaho.......................................................................................5

Table 2. Reduction error (RE) values for 10 cross sections resulting from the standard 

and genetic algorithm (GA)-reduction methods, Kootenai River, Idaho..............................9
iv

Conversion Factors 

[International System of Units to Inch/Pound] 

Multiply By To obtain

Length

meter (m) 3.281 foot (ft) 

Area

square meter (m2

) 10.76 square foot (ft2) 

Flow rate

cubic meter per second (m3

/s) 35.31 cubic foot per second (ft3/s)

Supplemental Information

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

River miles are used to measure horizontal distance along a river from a downstream 

confluence and to designate cross-section locations, which provided consistency with 

river-mile marker locations as designated by the U.S. Army Corps of Engineers, as shown on 

topographic maps, and as cited in earlier studies.

A cross section is a series of distance and elevation data point pairs that describe the channel 

shape perpendicular to the mean flow direction.

Distance is the length across the channel from the left-most data point while facing 

downstream.

Abbreviations

ABC area between the cross sections

DEMs digital elevation models

FEMA Federal Emergency Management Agency

GA Genetic Algorithm

GPS global positioning system

LiDAR light detection and ranging

m/m meters per meter

npr number of reduced data points

1-D one dimensional

RTK real-time kinematic

RE reduction error

Datum

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 

(NAVD88).
Introduction 1

Reducing Cross Sectional Data Using a Genetic Algorithm 

Method and Effects on Cross-Section Geometry and 

Steady-Flow Profiles

By Charles Berenbrock

Abstract

Reduction of cross-sectional data using a genetic 

algorithm method, and the effects of data reduction on channel geometry and steady-flow profiles, were analyzed. Two 

reduction methods─standard and genetic algorithms─were 

used to reduce cross-sectional data from the Kootenai River 

in northern Idaho. Cross sections that are representative of 

meander, straight, braided, and canyon reaches were used to 

evalutate the reduction methods. Visual and hydraulic analyses 

were used to assess the methods. The genetic algorithmreduced cross sections approximated the shape of the original 

cross sections better than the standard-reduced cross sections. 

A greater number of cross-sectional data points were needed 

for reduced cross sections in the straight reach, and even 

more in the braided reach, because a greater amount of data 

points are needed to adequately define cross sections that 

have greater topographic varability. For the genetic algorithmreduction method, about 40 data points were needed to 

adequately define the shape of a reduced cross section in the 

braided reach compared to 10 to 20 data points in the meander 

and canyon reaches. The standard-reduction method needed 

about 70 data points for the braided reach and more than 

30 points for the meander and canyon reaches. The genetic 

algorithm can effectively reduce data while staying within the 

threshold set by the maximum number of points to be included 

in the reduced dataset.

The effects of reduced cross-sectional data points on 

steady-flow profiles were also determined. Thirty-five cross 

sections of the original steady-flow model of the Kootenai 

River were used. These two methods were tested for all cross 

sections with each cross section resolution reduced to 10, 

20 and 30 data points, that is, six tests were completed for 

each of the thirty-five cross sections. Generally, differences 

from the original water-surface elevation were smaller as the 

number of data points in reduced cross sections increased, but 

this was not always the case, especially in the braided reach. 

Differences were smaller for reduced cross sections developed 

by the genetic algorithm method than the standard algorithm 

method.

Introduction

Cross-sectional data are used for many purposes, such 

as the investigations of flood plain delineation, flow patterns, 

shear stress, sediment mobility and transport, channel evolution, and aquatic habitat conditions. Accuracy of cross-section 

data is important because it could affect channel-geometry 

determinations and water-surface profile calculations. For 

example, the consequence of errors in the water-surface elevation has a major effect on computations of velocity, shear 

stress, and sediment transport. Water-surface profiles in many 

studies are computed by using one-dimensional (1-D) stepbackwater models such as HEC-RAS (Brunner, 2010), which 

uses the standard step method for steady flow. The standard 

step method uses the energy, continuity, and flow resistance 

(for example, Manning’s) equations between cross sections to 

compute the water-surface elevation and streamflow velocity 

(Chow, 1959). 

All models have a limit to the number of points allowed 

in a cross section. For example, the HEC-RAS version 4 

model has a 500-point limit. This limit might seem large 

enough, but when cross sections are computer generated from 

Light Detection and Ranging (LiDAR) or Digital Elevation 

Models (DEMs), or data are collected with equipment such 

as an echo sounder, the number of cross-sectional data points 

tends to be quite substantial. In an earlier study (Barton and 

others, 2004), for example, approximately 400 cross sections 

were surveyed on the Kootenai River in northern Idaho in 

order to understand the hydraulic characteristics of the river 

and to promote hydraulic conditions that improve spawning 

conditions for the endangered Kootenai River white sturgeon. 

The number of data points for each cross section ranged from 

about 500 to more than 2,500 points (Moran and Berenbrock, 

2003; Barton and others, 2004). Only a few cross sections had 

more than 2,000 data points. More than half of the surveyed 

cross sections were included in a HEC-RAS model of the 

Kootenai River (Berenbrock, 2005, 2006a), and most cross 

sections were reduced to less than 150 data points.
2 Reducing Cross Sectional Data Using a Genetic Algorithm Method and Effects on Cross-Section Geometry and Steady-Flow Profiles

Large datasets must be reduced to the meet the limitations of the programs being used. Reduced datasets improve 

run-time performance and facilitate data transmission and 

storage. Selecting the appropriate data points to keep from 

among the hundreds or thousands of data points can be both 

challenging and tedious. However, reducing the number of 

cross-sectional data points can result in significant changes 

to the reduced cross section, which could affect computed 

water-surface elevations, streamflow velocity, shear stress, 

and sediment transport. Considerable care must be taken when 

reducing data so that computed errors and uncertainties remain 

small or within acceptable limits. It is important to understand 

the effects that reduced cross sections can have on National 

Flood Insurance Program, flood-inundation, habitat, and 

sediment-transport studies. The Federal Emergency Management Agency (FEMA) has indicated that there is no point 

minimum—the number of data points for defining a cross section—as long as the actual shape of the cross section is well 

defined (Federal Emergency Management Agency, 2007).

Much research has been done on error and uncertainty 

analysis in surface-water hydraulics. Research has been carried out on determining the optimal spacing between cross 

sections (Samuels, 1989; Castellarin and others, 2009); developing cross sections from topographic maps, LiDAR, and 

DEM data (Burnham and Davis, 1990; Pasternack and others, 

2004; Cook and Merwade, 2009); and interpolating cross 

sections between known cross sections (Traver and Miller, 

1993). Travis and Lokey (1999) developed a method to reduce 

cross-section data to 100 data points, the maximum limit of 

the HEC-2 model. Berenbrock (2006b) developed a genetic 

algorithm (GA) computer program that reduces the number 

of data points in a cross section to any size. He compared the 

GA-reduced cross sections to cross sections developed by 

standard reduction methods—selecting every 10th, 20th, or 

nth point and omitting the rest—for the same number of data 

points in a cross section. Reduced cross sections developed 

from standard and GA methods were compared to the original 

cross-sectional data, and results showed that the GA method 

produced smaller differences from the original cross-sectional 

data than those obtained by using standard procedures. Unfortunately, no research to date has been done to determine the 

optimal number of points that are needed in a cross section or 

the effects on cross-section geometry and steady-flow profiles. The optimal number of points depends on the degree of 

topographic variability and the scale of topography that is of 

interest. It also depends on laws, regulations, or the requirements of the funding party—for example, FEMA (Federal 

Emergency Management Agency, 2007). 

The purpose of this report is to describe an application of a GA to the reduction of cross-sectional data points, 

demonstrate that the GA is a viable approach, and to evaluate 

the effects of reduced cross sections on channel geometry and 

steady-flow profiles. First, the study compared the accuracy 

of two reduction approaches, standard and genetic algorithm 

methods. Data from 10 cross sections covering 4 different 

channel types on a river were reduced by standard and genetic 

algorithm methods. These reduction methods were employed 

because the raw (original) data are preserved, not averaged, 

interpolated, or extrapolated. Second, the study identified the 

sources and spatial distribution of error in different channel 

types and determined the requisite sample size for different 

scales of resolution and application. Third, the study examined 

the effects of data reduction on steady-flow profiles. From 

these components, the amount of reduction can be tailored to 

the goals of an application.

Reduction Methods

There are many data-reduction methods available. For 

this study, only methods that preserve the original data were 

considered. The advantages of preserving the original data are 

that the original features, including vertical banks and discontinuities, are maintained. The original cross section—whether 

it contains 2,000 data points or 20 data points—is more accurate than anything generated in part from those data. 

Stream-channel cross-section data from the Kootenai 

River (Barton and others, 2004) were used in the data-point 

reduction methods. Data for the streambed part of these cross 

sections were collected by connecting continuous Real-Time 

Kinematic (RTK) Global Positioning System (GPS) equipment 

to an echo sounder, and bank data were collected by connecting a RTK GPS to a laser rangefinder equipped with an angle 

encoder (Moran and Berenbrock, 2003). Berenbrock (2006b) 

specifically used 10 cross sections from the Kootenai River to 

substantiate a genetic algorithm (GA) for data-point reduction. 

Data points from these 10 cross sections were also used in this 

study because the original data were still available. The cross 

sections are 107.658, 151.438, 152.019, 154.972, 163.027, 

185.394, 199.727, 212.227, 216.622, and 219.881, which are 

defined by a station number in river miles1

 that corresponds 

to its location on the river. The number of data points in these 

sections ranged from 497 in cross-section 152.019 to 2,521 

in cross-section 154.972. Original data from the other cross 

sections on the Kootenai River were not available. However, 

there were still enough data points in the reduced cross sections by Barton and others (2004)—usually more than 100 

data points per cross section, with some sections containing 

several hundred data points—for further reduction. For this 

study, cross-section data were reduced to as few as 10 data 

points per cross section.

1

River mile locations are based on Columbia Basin Inter-Agency 

Committee (1965) river-mile index for the Kootenai/Kootenay River. River 

mile 0 is at the confluence of the Kootenay River and Columbia River near 

Castlegar, British Columbia, Canada, and river mile 152 is upstream on the 

Kootenai River near Bonners Ferry, Idaho.
Reduction Methods 3

Standard Reduction

The standard-reduction method to reduce data points is 

to keep every 10th, 20th, 30th, or nth point and discard (omit) 

the rest. This procedure was used in this study because of the 

standard-reduction method’s simplicity, ease, and quickness. 

The value of the nth data point for each cross section was 

different because the total number of data points in the cross 

sections was different. For example, cross-section 152.019 has 

497 data points. If the number of points was reduced to 20, 

then every 26th point would be selected with 2 points remaining (1+(26×19)=495; then 497−495=2). The first data point 

in the cross section is always kept, so a value of 1 is added to 

the number of intervals (19 for this example). Fewer points 

exist along the banks than on the streambed for this dataset 

because the bank data were collected manually with a laser 

rangefinder. The remaining two points in this example were 

inserted into two different intervals that spanned the streambed. Thus, the size of those intervals was increased to 27. This 

procedure ensures that the last point in the cross section is kept 

and, for this example, the 497th point (last point) was kept 

(1+(26×17)+(27×2)=497).

Genetic Algorithm Reduction

Reducing the number of data points in a cross section is a 

non-linear combinatorial problem and, therefore, is well suited 

for heuristic algorithms such as genetic algorithm (GA). GAs 

mimic the natural seletion and survival of the fittest and are 

well suited for solving combinatorial optimization problems in 

which there is a large set of candidate solutions (Fisher, 2013). 

Koza (1992, p. 18) provides the following definition of a GA:

The genetic algorithm is a highly parallel mathematical algorithm that transforms a set (population) 

of individual mathematical objects (typically fixedlength character strings patterned after chromosome 

strings), each with an associated fitness value, into 

a new population (i.e., the next generation) using 

operations patterned after the Darwinian principle 

of reproduction and survival of the fittest and after 

naturally occurring genetic operations (notably 

sexual recombination).

In a GA, a population is represented by a number of 

individuals called genes (strings of chromosomes). Individuals are produced by ‘mating’ (crossover of chromosomes) two 

individuals together and ‘mutating’ a chromosome. The fittest 

individuals in the new population are selected to breed and 

mutate again, passing their genetic information to their children to create a newer population, and the least fit individuals 

are discarded. The newer population is then used in the next 

iteration of the algorithm. This process is repeated until a number of iterations has been reached or the maximum number of 

consecutive iterations without any improvement to the best fit 

individual is exceeded. Note that each individual is a solution 

to the problem. In essence, the GA represents an “intelligent” 

exploitation of the search space in a random fashion to solve a 

problem. A more complete discussion on genetic algorithms is 

given in Goldberg (1989), Grefenstette (1990), Davis (1991), 

and Mitchell (2002).

The GA developed by Berenbrock (2006b) was used 

in this study to reduce cross-section point data because it is 

easy to use, fast, and preserves the original data. The fitness 

function in Berenbrock’s (2006b) GA, however, is biased 

because the x-value (distance) for each data point does 

not contribute to the fitness function—only the y-values 

(elevation) do. The function does not account for the varying 

distances between the data points (irregularly spaced data) 

and, thus, data points need to be regularly spaced along a 

cross section. However, most cross sections are composed of 

irregularly spaced points. To account for irregularly spaced 

points, the fitness function was modified to calculate the area 

between the original and reduced cross sections—noted as the 

area between the cross sections (ABC). Thus, ABC accounts 

for the contribution from both x and y values. The value of 

ABC is always positive, regardless of how the cross sections 

cross one another. To solve this mathematically, the absolute 

value of ABC is employed and is denoted as |ABC|. The 

modified two-conditional fitness function is as follows:

i

n

j

j 1

n (inc plimit) 

j

j 1

ABC i

f(i)

ABC 1 10 oth

if inc plimit

erwise 

=

−

=

=         + ×

   

∑ ≤

∑

(1)

where

f(i) is the value of fitness for individual i;

n is the number of data points in the original 

dataset;

ABCj is the area between the cross sections, original 

and reduced, for trait j;

inci is the number of included data points in 

individual i; and

plimit or point limit is the maximum number of 

points to be included in the reduced 

dataset.

Each individual in the GA represents a reduced cross section, and the fitness of an individual (thus, a fitness of traits 

or data-point combinations) is represented as a value from 

the fitness function. The GA minimizes the fitness function, 

Minimize f(i), to identify the best-fit or optimal individual 

from all possible data-point combinations.

A sample calculation of fitness for a hypothetical cross 

section is shown in figure 1. Reduced cross sections are 

composed of included and excluded points. Included points 

are data points that are kept from the original cross section, 

and excluded points are data points that are discarded from 

the original cross section. For the reduced cross section shown 

in figure 1, the included data points are at points 1, 2, 5, 6, 

9, and 11, and excluded data points are at points 3, 4, 7, 8, 

and 10. The area between the cross-sections (ABC), original 
4 Reducing Cross Sectional Data Using a Genetic Algorithm Method and Effects on Cross-Section Geometry and Steady-Flow Profiles

10

j=1

1

2

3

4

5

6

7

8

9 10

11

ABC9 = 13.4m2

ABC8 = 34.3m2

ABC7 = 25.2m2

ABC6 = 0.5m2

ABC10 = 24.1m2

ABC5 = 0m2

ABC4 = 0.4m2

ABC3 = 40.3m2

ABC1 = 0m2

ABC2 = 53.8m2

f(i) = ∑ |ABCj| = 0+53.8+40.3+0.4+0+0.5+25.2+34.3+13.4+24.1 = 192.0 m2

f(i) is the value of fitness for individual i, the reduced hypothetical cross section

ABCj

 is the area between the hypothetical and reduced cross sections for point pairs

EXPLANATION

Hypothetical cross section

Reduced hypothetical cross section

5 Vertical line differentiating 

 neighboring ABCs

Cross-sectional data point and 

 identifier

0

10

20

30

40

50

0 10 20 30 40 50 60 70

Distance, in meters, from left bank

Elevation, in meters

sac14-0521_fig 01

Figure 1. Fitness calculation for a hypothetical cross section. Fitness is calculated according to equation 1 as shown at the top of the 

figure. m2

, square meters.

and reduced, is calculated for each closest pair of points. For 

the first pair of points 1–2, ABC is calculated to be 0 square 

meters (m2) because the data points for both cross sections are 

the same. For the second pair (2–3), ABC is calculated to be 

53.8 m2. For point pairs 3–4 and 4–5, ABC is calculated to be 

40.3 m2 and 0.4 m2, respectively. For point pairs 5–6, ABC is 

calculated to be 0 m2 because the data points for both cross 

sections are the same. For point pairs 6–7, 7–8, 8–9, 9–10, 

and 10–11, ABC is calculated to be 0.5 m2, 25.2 m2, 34.3 m2, 

13.4 m2, and 24.1 m2, respectively. The total ABC is 192.0 m2, 

which is the fitness value for this reduced hypothetical cross 

section. Fitness serves to aggregate the errors of an individual 

into a single measure. It is a good measure of accuracy, but 

only between other individuals in the population, as it is scale 

dependent.

To validate the modifications made to the GA, the 10 

cross sections that were used to validate the original GA 

(Berenbrock, 2006b) were used. The GA was run with the 

same parameter values as in the original GA. The best fitness 

results are shown in table 1. The reduction method used by 

Barton and others (2004), however, was based on selecting 

points every 1 to 2 meters (m), plus additional user-specified 

points to capture any important missing topography as determined from visual inspection (Berenbrock, 2006b, p. 388); 

this results in a variable number of reduced data points per 

cross section (table 1, column 3), rather than a fixed number 

of points, as described by the standard-reduction method. To 

compare these methods, the fitness for the best fit reduced 

cross sections from Barton and others (2004) and original GA 

(Berenbrock, 2006b) were recalculated by using equation 1 

and presented in table 1. 
Comparison of Reduction Results 5

Table 1. Comparison of best fitness between several reduction methods for 10 cross sections on the Kootenai River, Idaho.

[GA, genetic algorithm]

Cross 

section

1

Number of 

data points 

in original 

dataset

1

Number of 

reduced 

data points 

(npr)

Percentage 

reduction 

of data 

points

2

Best fitness, 

in square meters

GA method 

percentage lower than

3

Barton 

and others 

(2004) 

method

3

Original GA 

(Berenbrock, 

2006b) method

GA method

Barton 

and others 

(2004) 

method

Original GA

(Berenbrock, 

2006b) method

107.658 886 126 85.8 4.26 3.87 2.52 40.8 34.9

151.438 987 181 81.7 7.02 14.08 3.53 49.7 74.9

152.019 497 117 76.5 6.37 8.27 3.47 45.5 58.0

154.972 2,521 444 82.4 8.72 10.27 5.69 34.7 44.6

163.027 1,723 123 92.9 1.34 3.57 1.05 21.6 70.6

185.394 548 94 82.8 1.10 0.92 0.56 49.1 39.1

199.727 762 81 89.4 0.96 1.37 0.51 46.9 62.8

212.227 1,534 126 91.8 1.12 1.15 0.60 46.4 47.8

216.622 696 86 87.6 1.96 2.99 1.16 40.8 61.2

219.881 2,024 130 93.6 1.63 5.63 1.36 16.6 75.8

Average = 86.4 39.2 57.0

1

Barton and others (2004).

2

Calculated from equation 1, which is based on the area between the original and reduced cross sections (ABC).

3

To compare these methods, the fitness for each best fit reduced cross section from Barton and others (2004) and original GA (Berenbrock, 2006b) were 

recalculated using equation 1.

The best fitness values for the GA runs were consistently less (better performance, more accurate) than those 

from Barton and others (2004) and original GA (Berenbrock, 

2006b) methods. On average, the GA fitness was 39.2 percent 

and 57.0 percent less than Barton and others (2004) and 

original GA methods, respectively. These results indicate that 

the GA cross sections are significantly more (better) defined or 

representative of the original cross section than cross sections 

by the other two reduction methods. Note that the original GA 

fitness values were usually greater (lower performance) than 

Barton and others (2004) fitness values because the original 

GA fitness function was optimized for regularly spaced data, 

not irregularly spaced data as constitute most cross sections.

The GA-reduction method preserves the detailed 

character of the original cross section better than Barton and 

others (2004) and original GA (Berenbrock 2006b) methods. 

The GA cross sections were generally more defined where 

the original cross section had more topographic variability. 

The GA cross section matched the original cross section quite 

well, especially in the relatively smooth areas. The point 

density in the relatively smooth areas was far more reduced in 

the GA section than in Barton and others (2004) and original 

GA sections. Conversely, the point density in the relatively 

rough areas (topographic variability) was increased more in 

the GA section than in Barton and others (2004) and original 

GA sections. For the most part, the detailed character of the 

original cross section was better preserved by using the GA 

method than using the other two reduction methods. Therefore, the modification that was made to the GA method is the 

more appropriate genetic algorithm-reduction method and was 

used throughout this study.

Comparison of Reduction Results

To evaluate the effect on cross-section geometry and 

steady flow profiles, comparisons from the standard-reduction 

method and GA-reduction method were conducted. The 

comparisons included the 10 cross sections from the Kootenai 

River in table 1. The original cross sections were reduced 

in size to 10, 20, and 30 data points using both standardreduction and GA-reduction methods. Harrelson and others 

(1994) determined that at least 20 data points are needed 

in a cross section to accurately describe the character of 

the channel. The reduction to 30 data points was selected 

because a greater amount of data points are needed if the 

cross section is quite broad or complex, such as the case 

with braided channels. The reduction to 10 data points was 

selected because it is one-half of the recommended minimum 

(Harrelson and others, 1994). Results from the reductions’ 

visual and hydraulic analyses are presented in the following 

sections. The practical consequences from both reduction 

methods are investigated in the “Hydraulic Modeling Analysis 

of Reduction Methods” section by the use of one-dimensional 

(1-D) steady-flow profiles.
6 Reducing Cross Sectional Data Using a Genetic Algorithm Method and Effects on Cross-Section Geometry and Steady-Flow Profiles

Visual Analysis of Cross-Section Reductions

After reducing the cross-section data, the reduced 

datasets were viewed graphically and compared to the original 

data. Although the analysis of fitness is useful as given in 

table 1, it is a black-box approach that reports aggregate 

results without providing an understanding of the spatial 

details and does not distinguish which parts of the cross section are causing the error. Visual analysis provides further 

insight regarding the spatial distribution of error for the two 

reduction methods.

Synder and Minshall (1996) identified three geomorphic 

reaches in the Kootenai River—a meander reach, a braided 

reach, and a canyon reach. Barton and others (2005) defined a 

fourth geomorphic reach—a straight reach. The meander reach 

is a single channel with gentle bends. The streambed consists 

primarily of fine sand. Water depths usually exceed 12 m, 

and the water-surface slope is about 2×10-5 meters per meter 

(m/m), less than one-twentieth the slope in the braided reach. 

Sand dunes—as high as 1.4 m and as long as about 23 m 

(Barton and others, 2005)—also occur throughout the meander 

reach. The straight reach is a transitional reach between the 

meander and braided reaches, and its streambed consists primarily of sand, gravel, and cobbles. The braided reach usually 

consists of multiple channels, and the streambed is composed 

primarily of gravel and cobbles. Water depths usually are less 

than 2 m, and water-surface slope is about 4.6×10-4 m/m. The 

canyon reach consists of a long, straight single channel with 

steep canyon walls and is incised into bedrock. The streambed 

consists primarily of cobbles and boulders. Water depths are 

usually about 6 m, and water-surface slope is about 3×10-4

m/m.

Examples for each geomorphic or channel type—

meander, straight, braided, and canyon (Czuba and Barton, 

2011)—are shown in figure 2. Cross-section 107.658 is in 

the meander reach, cross-section 152.019 is in a straight 

reach, cross-section 154.972 is in a braided reach, and crosssection 163.027 is in a canyon reach. The effects of standard 

data-point reduction on cross-sectional shape are shown in 

figures 2A, 2C, 2E, and 2G, and effects of GA reduction on 

cross-sectional shape are shown in figures 2B, 2D, 2F, and 2H.

At first glance, the reduced cross sections composed of 

10 data points showed that it was the worst shaped case for 

both reduction methods. The graphs in figure 2 indicate that 

more than 30 points were needed when using the standardreduction method, whereas 20 to 30 data points were adequate 

when using the GA-reduction method, except for cross-section 

154.972 (fig. 2D). The GA-reduced cross sections approximated the shape of the original cross sections better than 

the standard-reduced cross sections for the same number of 

reduced data points (npr). The standard-reduction method produced greater bank errors than the GA method; also there were 

greater streambed errors in the braided and straight reaches. 

At cross-section 154.972 (braided reach), a greater number of 

data points were needed for both reduction methods because 

there was much more topographic variability in this cross 

section than in the other cross sections. Additional analysis 

indicated (not shown in fig. 2) that about 70 data points were 

needed to adequately define the shape of cross-section 154.972 

when using the standard-reduction method and about 40 points 

when using the GA-reduction method.

A visual comparison was performed on cross-sectional 

area plots. Area was used instead of conveyance because 

the performance of conveyance can be confounded by the 

uncertainly in coefficients in Manning’s flow equation 

(Chow, 1959). The effects of data-point reduction on crosssectional area for the cross sections in figure 2 are shown 

in figure 3. The cross-sectional area curves for the reduced 

cross sections composed of 10 points were furthest from the 

original area curves for both reduction methods, and were 

closest to the original curves for the 30-point cross sections 

for both methods. Cross-sectional areas for the GA-reduced 

cross sections approximated the shape of the original crosssectional areas better than the standard-reduced cross sections 

for respective data-point reduction. But for cross-section 

154.972 (braided reach), none of the reduced area curves for 

both methods closely agreed with the original area curves 

(figs. 3E, F). The original area curves show a break near 

an elevation of 534 m that is probably caused by a greater 

amount of topographic variability in the original cross section 

(figs. 2E, F) compared to the reduced cross sections. For the 

reduced cross sections, this departure at the break in slope 

indicates that there are not enough data points to adequately 

define the topographic variability in this area.

The performance of the cross-section reduction is 

quantified by the measured error in the reduced cross section. 

This error is designated as reduction error (RE) and is defined 

as the total area between the original and reduced crosssection curves (ABC) divided by the number of reduced data 

points (npr) and is expressed as follows:

ABC RE=

npr

(2)

The RE normalizes the reduction methods with respect to 

the number of npr and allows for direct comparison of different cross sections and reduction methods. The RE has units of 

area per point; for the datasets used in this study, it was square 

meters per reduced data point (m2

/point). The value for RE 

depends on the number of points in the original and reduced 

cross sections and the topography of the cross section. The 

smaller the RE, the greater similarity between the original 

and reduced cross sections. The larger the RE, the greater the 

disparity between the topography of the original and reduced 

cross sections. Equation 2 represents the gain in crosssectional likeness due to the reduction of the area between 

the cross-section curves with the increase in the number of 

reduced data points. The RE values are unique and depend 

on cross-section topography and the number of data points in 

the original and reduced cross sections. The accuracy of the 

cross-section reduction is likely to be sensitive to the quality 

and quantity of the original data and how it represents the 

different scales of topography that are present. For example, 
Comparison of Reduction Results 7

520

525

530

535

540

545

525

530

535

540

545

Genetic algorithm (GA) reduction

525

530

535

540

545

Standard reduction 

Standard reduction 

532

534

536

538

540

542

Standard reduction 

Standard reduction GA reduction 

GA reduction 

GA reduction 

Distance, in meters, from left bank 

538

540

542

544

546

548

550

538

540

542

544

546

548

550

EXPLANATION

Original

10 data points

20 data points

30 data points

Cross sections

A. Cross section 107.658—Meander Reach 

C. Cross section 152.019—Straight Reach 

E. Cross section 154.972—Braided Reach 

G. Cross section 163.027—Canyon Reach H. Cross section 163.027—Canyon Reach 

F. Cross section 154.972—Braided Reach 

D. Cross section 152.019—Straight Reach 

B. Cross section 107.658—Meander Reach 

520

525

0

0 50 100 200 300

0

0 50 100 150 0 50 100 150

200 400 600 800 0 200 400 600 800

150 250 0 50 100 150 200 250 300

50 100 150 200 0 50 100 150 200

530

535

540

545

Elevation, in meters

532

534

536

538

540

542

sac14-0521_fig 02

Figure 2. Effects of data-point reduction on cross-sectional shape. A, Cross-section 107.658, meander reach, standard reduction. B, 

Cross-section 107.658, meander reach, genetic algorithm (GA) reduction. C, Cross-section 152.019, straight reach, standard reduction. 

D, Cross-section 152.019, straight reach, GA reduction. E, Cross section 154.972, braided reach, standard reduction. F, Cross section 

154.972, braided reach, GA reduction. G, Cross section 163.027, canyon reach, standard reduction. H, Cross section 163.027, canyon 

reach, GA reduction.
8 Reducing Cross Sectional Data Using a Genetic Algorithm Method and Effects on Cross-Section Geometry and Steady-Flow Profiles

EXPLANATION

Original

10 data points

20 data points

30 data points

Cross sections

A. Cross section 107.658—Meander Reach 

C. Cross section 152.019—Straight Reach 

E. Cross section 154.972—Braided Reach 

G. Cross section 163.027—Canyon Reach H. Cross section 163.027—Canyon Reach 

D. Cross section 152.019—Straight Reach 

B. Cross section 107.658—Meander Reach 

520

2,500

2,000

1,500

1,000

3,000

5,000

4,000

3,000

2,000

1,000

0

2,000

1,000

2,500

500

0

1,500

3,000

2,000

1,000

2,500

500

0

1,500

500

0

525 530 535 540 545

Cross-sectional area, in square meters

520

2,500

2,000

1,500

1,000

500

0

525 530 535 540 545

Standard reduction 

520 525 530 535 540 545

530 532 534 536 538 540 542 530 532 534 536 538 540 542

Standard reduction 

1,400

1,200

1,000

800

200

400

600

0

538 540 542 544 546 548 550 552

Standard reduction 

1,400

1,200

1,000

800

200

400

600

0

538 540 542 544 546

Elevation, in meters

548 550 552

GA reduction 

F. Cross section 154.972—Braided Reach 5,000

4,000

3,000

2,000

1,000

0

GA reduction 

GA reduction 

520 525 530 535 540 545

Standard reduction Genetic algorithm (GA) reduction

sac14-0521_fig 03

Figure 3. Effects of data-point reduction on cross-sectional area. A, Cross-section 107.658, meander reach, standard reduction. B, 

Cross-section 107.658, meander reach, genetic algorithm (GA) reduction. C, cross-section 152.019, straight reach, standard reduction. 

D, Cross-section 152.019, straight reach, GA reduction. E, cross section 154.972, braided reach, standard reduction. F, Cross section 

154.972, braided reach, GA reduction. G, Cross section 163.027, canyon reach, standard reduction. H, Cross section 163.027, canyon 

reach, GA reduction.
Comparison of Reduction Results 9

the performance of the cross-section reduction could differ 

if a topographically complex braided channel was originally 

surveyed by 40 data points as opposed to 400 data points. This 

is not an issue for the Kootenai data, given the high density of 

original data points, but it could be an issue in other studies.

Values of RE for reduced cross sections that contained 

10, 20, and 30 data points were calculated for the 10 cross 

sections on the Kootenai River by using the standard- and GAreduction methods (table 2). For the GA-reduced cross sections, fitness or ABC was calculated by the GA program. For 

the standard-reduced cross sections, ABC was calculated using 

the first part of equation 1, similar to the calculation shown 

in figure 1. Then, equation 2 was used to calculate the RE for 

both methods. As shown in table 2, as the number of reduced 

data points increase, the RE decreases. Also the RE values for 

the GA-reduced cross sections were always less than the RE 

values for the standard-reduced cross sections, indicating that 

the GA-reduced cross sections are more representative of the 

original data than the standard-reduced cross sections. The 

RE values for the canyon-reach cross sections were usually 

consistent with one another for the number of data points for 

both methods. For the GA method, the canyon cross sections 

had the lowest RE values, and cross-section 154.972 (braided 

reach) had the highest RE values. For the standard-reduction 

method, cross-section 107.658 (meander reach) had the lowest 

RE values because the banks in this cross section were gently 

sloping, thus, reducing its ABC value (fig. 2A). Also, for the 

standard-reduction methods, cross-section 152.019 (straight 

reach) had the highest RE values because the method selected 

only a few points on the banks, thus, causing ABC to be quite 

large (fig. 2C). Similarly, that is the reason for the large RE 

values for cross-section 151.438 (straight reach).

The GA program was used to develop RE curves (fig. 4) 

for the 10 cross sections listed in table 2. Each curve was 

developed by running the program for a selected number of 

reduced data points (npr) starting at 10 data points and incrementing by 10 until reaching 100 data points. The program 

was run 10 times at every npr to ensure that near optimal 

results were reached. The lowest RE value (calculated from 

equation 2) at each npr was retained and used to develop 

the RE curve for every cross section. Also, the ABC and npr 

values from table 1 for the modified GA were used, and, for 

seven cross sections, they were used to extend the RE curves 

beyond an npr of 100. These curves represent near optimal 

solutions. All RE curves in figure 4 are concave upward to the 

right. The RE curve for cross-section 154.972 (braided reach) 

is more upward toward the right than all the other RE curves 

because its cross section was more complex (greater topographic variability). In contrast, cross sections in the meander 

and canyon reaches were less complex (less topography 

variability), causing the RE curves to be in the lower part of 

the plot. The RE curves for cross sections in the straight reach 

were between the canyon and braided curves. The RE plot 

(fig. 4) shows that as topographic variability in a cross section 

increases, the RE curve will be more upward toward the right 

in the plot, thereby indicating that increasing the number of 

points in a cross section needs to be increased to adequately 

represent the original cross section.

The RE curve also allows one to visually judge where 

an increase in npr does not result in significant RE reduction 

(called the point of diminishing returns). This location on a RE 

curve is at the point of diminishing returns (breakpoint). For 

this study, the breakpoint’s location was determined by a twophase linear regression where two straight lines are fitted to 

the data by minimizing the residual sum of squares. Above the 

breakpoint (to the left on the curve), the RE value increases 

quite rapidly as the number of data points decrease; below 

the breakpoint (to the right on the curve), the increase in the 

number of data points does not lower the RE value as rapidly. 

The break points for the 10 RE curves are shown in figure 4. 

The location of the breakpoint for cross-section 107.658 

(meander reach) is at an RE of about 0.4 m2

/point and at a 

number of reduced data points of 22 points, about 1 m2/point 

and 22 points for cross-section 152.019 (straight reach), about 

3 m2/point and 29 points for cross-section 154.972 (braided 

reach), and about 0.4 m2/point and 23 points for cross-section 

163.027 (canyon reach). Breakpoints (points of diminishing 

returns) for cross sections in the meander and canyon reaches 

were less than those in the straight and braided reaches 

because their topography is less varied (fig. 4). By 50 data 

Table 2. Reduction error (RE) values for 10 cross sections 

resulting from the standard and genetic algorithm (GA)-reduction 

methods, Kootenai River, Idaho.

Cross 

section

Reduction error, 

in square meters per reduced point

Number of reduced data points in cross section

Standard method GA method

10 20 30 10 20 30

Meander Reach

107.658 4.9 1.7 0.8 2.0 0.6 0.2

Straight Reach

151.438 33.9 9.0 4.3 11.6 2.1 0.9

152.019 49.8 18.1 10.4 10.9 1.7 0.6

Braided Reach

154.972 41.1 10.4 4.2 18.2 4.7 2.6

Canyon Reach

163.027 11.8 6.1 3.9 3.5 0.7 0.1

185.394 13.0 3.4 1.5 1.8 0.2 0.1

199.727 11.1 4.0 2.5 3.4 0.1 <0.1

212.227 9.7 2.5 1.5 4.1 0.9 0.2

216.622 7.6 2.7 1.3 3.5 0.6 0.2

219.881 17.0 4.6 2.1 2.7 1.2 0.6
10 Reducing Cross Sectional Data Using a Genetic Algorithm Method and Effects on Cross-Section Geometry and Steady-Flow Profiles

EXPLANATION

Cross section reach types with cross- 

 section numbers in parenthesis

Meander (107.685)

Straight (151.438, 152.019)

Braided (154.972)

Canyon (163.027, 185.394, 199.727,

 212.227, 216.622, 219.881)

Reduction error

Breakpoint–point of diminishing returns.

sac14-0521_fig 04 Reduction error (RE), in square meters per reduced data point

Number of reduced data points in cross section

RE = 0.25

Increasing

topography

variability

0 20 40 60 80 100 120

0

2

4

6

8

10

12

14

16

18

20

Reduction error (RE), in square meters per reduced data point

Increasing

topography

variability

140

Number of reduced data points in cross section

10 15 20 25 30 35 40

0

1

2

3

4

Figure 4. Reduction-error (RE) curves for 10 cross sections resulting from the genetic algorithm (GA)-reduction method, Kootenai 

River, Idaho.
Comparison of Reduction Results 11

points, all cross sections except 154.972 (braided reach) had 

RE values equal to or less than 0.25 m2/point (fig. 4). At this 

value, the RE curves for the meander, straight, and canyon 

cross sections are nearly asymptotic to the x-axis. Crosssection 154.972 (braided reach) did not reach an RE value of 

0.25 m2/point until 93 data points. This again indicates that 

cross sections having varied topography such as the braided 

reach, require more data points to define them adequately.

Hydraulic Modeling Analysis of Reduction 

Methods

The effects of cross-sectional data-point reduction along 

a reach were examined on steady-flow water-surface profiles. 

The 1-D hydraulic-flow model of the Kootenai River in 

Idaho (Berenbrock, 2005, 2006a) was used to evaluate these 

effects. The HEC-RAS model, version 4.1 (Brunner, 2010), 

was used to compute the steady flow profiles (water-surface 

elevations at cross sections). Only part of the original model 

(164 cross sections) was used. Cross-sectional data points 

for only 35 cross sections were reduced—starting at crosssection 149.910 (meander reach) and stopping at cross-section 

156.861 (braided reach). This reach was selected because it 

is the focus of a habitat-restoration project for the recovery 

of the endangered Kootenai River white sturgeon (Acipenser 

transmontanus) population, and misrepresentation of crosssectional data could have major effects on computed steadyflow profiles. This reach included meander, straight, and 

braided reaches. The model was not extended into the canyon 

reach because reduction in cross-sectional data points had little 

effect on the computed cross sections, as shown by the low RE 

values (table 2, fig. 4). The transition from meander to straight 

occurs near river mile 151, straight to braided at river mile 

153.3, and braided to canyon near river mile 161. A total of 30 

simulations were run using the combinations of 5 discharges, 

3 data-point reduction levels, and 2 reduction methods. The 

five discharges (170, 283, 850, 1,416, and 1,982 cubic meters 

per second, or m3

/s) represent the objective discharges, with 

respect to habitat restoration (Berenbrock, 2006a), and cross 

sections were reduced to 10, 20, and 30 data points because 

they span the breakpoint values for cross sections in figure 4. 

The two data-point reduction methods used were standard and 

GA.

Results for water-surface elevations are given as differences from the original (fig. 5). Generally, results from the 

simulations showed that the standard-reduced cross sections 

had greater water-surface differences from the original than 

did simulations from GA-reduced cross sections. Also differences were greater for the 10-point simulations than for the 

20-point and 30-point simulations in respective methods and 

discharges. For all simulations, the greatest water-surface elevation differences were at cross sections in the braided reach, 

probably because the reduced cross sections in that reach were 

not as accurate in representing the original, as shown by the 

higher RE values. For the standard-reduction simulations, 

effects from the reduced cross section were seen upstream of 

river mile 161 (not shown on fig. 5), and for the GA-reduced 

simulations, no effects were seen upstream of river mile 159 

(fig. 5). In simulations where the RE value for every cross 

section was equal to or less than 0.25 m2

/point, differences in 

water-surface elevations (steady flow profile) from the original 

were very small.

Some cross sections had greater differences in watersurface elevation when more data points were used (fig. 5). 

This is contrary to the paradigm that more data are better. 

For example, as seen in figure 5A, model results for the 

standard-reduction method showed that water-surface differences for the 20-point simulation were less than for the 

30-point simulation. This occurred in and around cross-section 

154.575 (braided reach). The error depends on which crosssectional data points are captured during reduction and their 

importance in defining the cross section. The original cross 

section of 154.575 has four braided channels—three shallow 

secondary channels and one deep main channel—when total 

discharge is 170 m3/s (fig. 6). The left most secondary channel was characterized by 6 and 10 data points in the 20-point 

and 30-point standard-reduced cross sections, respectively 

(fig. 6A). However, water-surface differences were less for the 

20-point simulation than for the 30-point simulation (fig. 5A). 

This also occurred at other cross sections, at other discharges, 

and for both reduction methods. Figure 6A shows that both 

point-reduction levels poorly fit the braided bar landforms in 

the middle of the cross section, but both point-reduction levels 

fit the original quite well in the deep main channel (right). The 

braided bar landforms were reduced in size in the 20-point 

and 30-point standard reduced cross sections. For the 10-point 

standard reduced cross section, only two channels were seen—

a shallow and wide secondary channel (left), and one deep 

main channel (right) in which the braided bar landforms from 

the original cross section are missing.

At discharges of 850 m3/s and greater, model results for 

the standard-reduction method showed differences in watersurface elevation were less for the 20-point simulations than 

for the 30-point simulations in the braided reach, specifically 

in and around cross-section 156.604 (fig. 5C–E). At this cross 

section, the 20-point and 30-point standard-reduced cross 

sections excluded data points in the thalweg of the secondary 

channel, which caused the elevation of the reduced cross section in this area to be 0.5 m higher than the original. However, 

the 10-point standard-reduced cross section included it, but 

excluded other secondary thalweg points in the cross section. 

Even though the 30-point standard-reduced cross sections 

had more points than the 20-point cross section, the 20-point 

standard-reduced cross section had points in locations that 

represent the cross-section topography more accurately.

Generally, the GA-reduced cross sections incorporated 

the shallow channel thalweg and better represented the 

original cross section. However, at discharges of 170 and 

283 m3

/s, water-surface differences for the 10-point GAreduced simulation were less than the 20-point simulation in 

and around cross-section 154.575 (fig. 5A–B). Although the 
12 Reducing Cross Sectional Data Using a Genetic Algorithm Method and Effects on Cross-Section Geometry and Steady-Flow Profiles

EXPLANATION 

Genetic algorithm (GA) reduction for reduction 

error (RE) ≤ 0.25 square meters per point

30-data-point GA reduction

20-data-point GA reduction 

10-data-point GA reduction 

30-data-point standard reduction

20-data-point standard reduction

10-data-point standard reduction

River miles

River miles

Differences in water-surface elevation, in meters, from original 

Meander

reach

Straight

reach

Braided reach

Meander

reach

Straight

reach

Braided reach Meander

reach

Straight

reach

Braided reach

Meander

reach

Straight

reach

Braided reach Meander

reach

Straight

reach

Braided reach

149 150 151 152 153 154 155 156 157 158 159 160

149 150 151 152 153 154 155 156 157 158 159 160 149 150 151 152 153 154 155 156 157 158 159 160

149 150 151 152 153 154 155 156 157 158 159 160

A. Discharge = 170 cubic meters per second

E. Discharge = 1,982 cubic meters per second

C. Discharge = 850 cubic meters per second

B. Discharge = 283 cubic meters per second

D. Discharge = 1,416 cubic meters per second

-0.4

-0.3

-0.2

-0.1

0.1

0.2

0.3

0

-0.4

-0.3

-0.2

-0.1

0.1

0.2

0.3

0

-0.4

-0.3

-0.2

-0.1

0.1

0.2

0.3

0

149 150 151 152 153 154 155 156 157 158 159 160

sac14-0521_fig 05

Figure 5. Effects of reduced cross sections on simulated water-surface elevation at five river discharges. A, Discharge of 170 cubic 

meters per second (m3

/s). B, Discharge of 283 m3/s. C, Discharge of 850 m3/s. D, Discharge of 1,416 m3/s. E, Discharge of 1,982 m3/s. 

≤, less than or equal to. 
Comparison of Reduction Results 13

Q = 850 m3/s

Q = 283 m3

/s

Q = 170 m3

/s

0 100 200 300 400 500 600

530

532

534

536

538

540

542

544

EXPLANATION

Original

30 data point

20 data point

10 data point

A. Standard reduction method

Elevation, in meters

Q = 850 cubic meters per second (m3/s)

Q = 283 m3

/s

Q = 170 m3

/s

Water surface

Cross sections

EXPLANATION

Original

30 data point

20 data point

10 data point

Water surface

Cross sections

530

532

534

536

538

540

542

544

B. Genetic algorithm (GA) reduction method

0 100 200 300 400 500 600

Elevation, in meters

Distance, in meters, from left bank

sac14-0521_fig 06

Figure 6. Comparisons between the original and 10 point, 20 point, and 30 point reduced cross sections produced by two reduction 

methods for cross-section 154.575. A, Standard reduction method. B, Genetic algorithm (GA) method.
14 Reducing Cross Sectional Data Using a Genetic Algorithm Method and Effects on Cross-Section Geometry and Steady-Flow Profiles

20-point reduced cross section resembled the original cross 

section better than the 10-point reduced cross section (fig. 6B), 

it did worse when it came to step-backwater analysis because 

the water-surface elevation at cross-section 154.178, a reduced 

cross section just downstream of this section, contained large 

errors from the original. At cross-section 154.178, there were 

too few data points in the secondary channels to define the 

topographic variability adequately, especially the thalwegs in 

the secondary channels.

Differences from the original water-surface elevation in 

the steady-flow profiles were small in the GA-reduced cross 

sections when an RE of less than or equal to 0.25 m2/point was 

used (fig. 5). The number of reduced data points for this condition ranged from 20 to 40 points, but several cross sections 

in the braided reach contained more. Cross-section 154.972 

(braided reach), for example, had the most reduced data points 

(93) in order to meet this condition. This value is less than 

100, the limit of most hydraulic and sediment transport models 

such as HEC-2 and HEC-6. Note that the level of acceptable 

error in the water-surface elevation or flow depth depends on 

one’s intended use (for example, floodplain maps, sediment 

transport, or fish habitat).

Summary

The genetic algorithm (GA) method is a viable approach 

for reducing data points in a cross section and produced better 

results than the standard method it was tested against in this 

study. The original GA by Berenbrock (2006b) did not account 

for varying distances between cross-sectional data points. To 

account for irregularly spaced data points, the fitness function 

was modified to calculate the area between the original and 

reduced-cross-section curves. By using 10 cross sections from 

the Kootenai River, best fitness values were consistently lower 

(demonstrating better performance) for the GA runs than for 

the standard-method and original GA runs. On average, the 

GA fitness was 39.2 percent lower than the standard method, 

and for several cross sections was nearly 50 percent lower. 

The GA-reduced cross sections approximated the shape of 

the original cross sections better than the standard method 

and, thus, the GA-reduction method should be used over the 

standard method.

To provide further insight regarding the spatial 

distribution of error for the two approaches, visual and 

hydraulic analyses were completed. Visual analysis (graphs) 

demonstrated that GA-reduced cross sections approximated the shape of the original cross section better than the 

standard-reduced cross sections. This was also true for the 

cross-sectional area. An reduction error (RE) was developed to 

quantify the difference between the original and reduced cross 

sections. The RE values decreased as the number of reduced 

data points increased for both reduction methods, and as 

expected, RE values were lower (better) for the GA-reduced 

cross sections than for the standard method. The RE curves 

were developed for the 10 cross sections on the Kootenai 

River by using the GA-reduction method, and the breakpoints (points of diminishing returns) found. For the canyon 

and meander reaches, the breakpoints (about 20 data points) 

represent the optimal number of points needed in a cross section. However, many more cross-sectional data points were 

needed for cross sections in the braided and straight reaches 

as compared to cross sections in the canyon and meander 

reaches. Also, additional cross sections from other study 

areas are needed to draw consistent conclusions regarding the 

number of cross-sectional data points needed for each reach 

type. The GA-reduced cross sections matched the shape of the 

original cross section quite well when the RE was equal or less 

than 0.25 m2

/point, the point at which the RE curves become 

approximately asymptotic to the x-axis. Most cross sections 

reached this value at 20 to 40 data points, but cross-section 

154.972 (braided reach) did not reach it until 93 data points. 

More complexly shaped cross sections need greater amounts 

of data points to define them adequately. Depending on the 

intended use of a cross section, the number of data points 

depends on the degree of topographic variability of the cross 

section and the scale of interest.

This study also investigated the practical consequences 

of errors due to cross-section reduction on steady-flow 

profiles. Thirty-five cross sections from the original steadyflow surface-water model of the Kootenai River were used. 

Cross-sectional data in these cross sections were reduced 

to 10, 20, and 30 data points for both reduction methods. 

Results generally indicated that differences were less for 

cross sections developed by the GA-reduction method than 

by the standard-reduction method. Also, differences from 

the original water-surface elevation were usually less as the 

number of data points in cross sections increased—except for 

some of the reduced cross sections in the braided and straight 

reaches. The exception is contrary to the paradigm that more 

data points are better, and is the result of the standard and GA 

methods not always having enough points in the secondary 

channels (braided) to define them adequately. The GA method 

did not select enough points in the secondary channels because 

fitness was not bettered (lower value) by doing so. To rectify 

this problem, the GA needs to be modified so that thalweg 

points in all channels, the main as well as secondary, are 

selected. Although the GA method is clearly a major advancement in the reduction of cross-sectional data, there are, of 

course, limits to its performance. In particular, cross sections 

having multiple channels, such as braided channels, can be 

problematic.
References Cited 15

References Cited

Barton, G.J., McDonald, R.R., Nelson, J.M., and Dinehart, 

R.L., 2005, Simulation of flow and sediment mobility using 

a multidimensional flow model for the white sturgeon 

critical-habitat reach, Kootenai River near Bonners Ferry, 

Idaho: U.S. Geological Survey Scientific Investigations 

Report 2005–5230, 54 p.

Barton, G.J., Moran, E.H., and Berenbrock, Charles, 2004, 

Surveying cross sections of Kootenai River between Libby 

Dam, Montana, and Kootenay Lake, British Columbia, 

Canada: U.S. Geological Survey Open-File Report 2004–

1045, 35 p.

Berenbrock, Charles, 2005, Simulations of hydraulic characteristics in the white sturgeon spawning habitat of the 

Kootenai River near Bonners Ferry, Idaho: U.S. Geological 

Survey Scientific Investigations Report 2005–5110, 30 p.

Berenbrock, Charles, 2006a, Simulations of hydraulic characteristics for an upstream extension of the white sturgeon 

habitat of the Kootenai River near Bonners Ferry, Idaho—A 

supplement to Scientific Investigations Report 2005–5110: 

U.S. Geological Survey Scientific Investigations Report 

2006–5019, 17 p.

Berenbrock, Charles, 2006b, A genetic algorithm to reduce 

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16 Reducing Cross Sectional Data Using a Genetic Algorithm Method and Effects on Cross-Section Geometry and Steady-Flow Profiles

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Publishing support provided by the U.S. Geological Survey 

Science Publishing Network, Sacramento Publishing Service Center

For more information concerning the research in this report, contact the

 Director, California Water Science Center

U.S. Geological Survey

6000 J Street, Placer Hall

Sacramento, California 95819

http://ca.water.usgs.gov
Charles Berenbrock—Reducing Cross-Sectional Data Using a Genetic Algorithm Method and Effects on Cross-Section Geometry and Steady-Flow Profiles—

Scientific Investigations Report 2015–5034

ISSN 2328-0328 (online)

http://dx.doi.org/10.3133/sir20155034