A Method for Determining Average Beach Slope and Beach 

Slope Variability for U.S. Sandy Coastlines

By Kara S. Doran, Joseph W. Long, and Jacquelyn R. Overbeck

Open-File Report 2015-1053

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 

For more information on the USGS—the Federal source for science about the Earth, 

its natural and living resources, natural hazards, and the environment—visit 

http://www.usgs.gov or call 1–888–ASK–USGS

For an overview of USGS information products, including maps, imagery, and publications, 

visit http://www.usgs.gov/pubprod

To order this and other USGS information products, visit http://store.usgs.gov

Suggested citation: 

Doran, K.S., Long, J.W., and Overbeck, J.R., 2015, A method for determining average beach slope and beach 

slope variability for U.S. sandy coastlines: U.S. Geological Survey Open-File Report 2015-1053, 5 p., 

http://dx.doi.org/10.3133/ofr20151053.

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

endorsement by the U.S. Government.

Although this report is in the public domain, permission must be secured from the individual 

copyright owners to reproduce any copyrighted material contained within this report.

ISSN 2331-1258 (online)

ii
iii

Contents

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

Data Acquisition ....................................................................................................................................................... 1

Data Processing ....................................................................................................................................................... 1

Uncertainty Analysis ................................................................................................................................................. 2

References Cited...................................................................................................................................................... 5

Figures

1.Cross-shore profile of lidar-based elevations from the Outer Banks of North Carolina, surveyed in November 

2012, indicating the locations and elevations of the dune crest, dune toe, shoreline, mean beach slope, 

mean-high-water, and high water line. ........................................................................................................... 2

2. An example of spatially and temporally averaged beach slopes computed using data collected 

between 1996 and 2012 for the Outer Banks of North Carolina. ................................................................... 3

3. Spatially and temporally averaged beach slopes for a 1-kilometer section of coast in the Outer Banks of 

North Carolina ................................................................................................................................................ 4
iv

Conversion Factors

SI to Inch/Pound

Multiply By To obtain

Length

centimeter (cm) 0.3937 inch (in.)

meter (m) 3.281 foot (ft) 

meter (m) 1.094 yard (yd) 

kilometer (km) 0.6214 mile (mi)

Datum

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 (NAVD 88).

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).

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

Abbreviations

HWL High water line

MHW Mean high water

MRE Mean residual error

NOAA National Oceanic and Atmospheric Administration

RMS Root mean square

USACE U.S. Army Corps of Engineers

USGS U.S. Geological Survey
1

A Method for Determining Average Beach Slope and Beach 

Slope Variability for U.S. Sandy Coastlines

By Kara S. Doran, Joseph W. Long, and Jacquelyn R. Overbeck

Introduction 

The U.S. Geological Survey (USGS) National Assessment of Hurricane-Induced Coastal Erosion 

Hazards compares measurements of beach morphology with storm-induced total water levels to produce forecasts of coastal change for storms impacting the Gulf of Mexico and Atlantic coastlines of 

the United States. The wave-induced water level component (wave setup and swash) is estimated by 

using modeled offshore wave height and period and measured beach slope (from dune toe to shoreline) 

through the empirical parameterization of Stockdon and others (2006). Spatial and temporal variability in beach slope leads to corresponding variability in predicted wave setup and swash. For instance, 

seasonal and storm-induced changes in beach slope can lead to differences on the order of 1 meter (m) 

in wave-induced water level elevation, making accurate specification of this parameter and its associated 

uncertainty essential to skillful forecasts of coastal change. A method for calculating spatially and temporally averaged beach slopes is presented here along with a method for determining total uncertainty 

for each 200-m alongshore section of coastline.

Data Acquisition 

The combination of laser-based ranging with inertial and GPS-based navigation provides an 

efficient method for collecting high-resolution data of sub-aerial topography with sufficient accuracy 

(root-mean-square [RMS] vertical accuracy = 15 centimeters [cm]; horizontal accuracy = 1–1.5 m) to 

resolve the spatial details of coastal morphology (Sallenger and others, 2003). Processed lidar datasets 

have been acquired from various sources including the USGS, U.S. Army Corps of Engineers (USACE), 

and the National Oceanic and Atmospheric Administration (NOAA) Digital Coast Web site. Lidar datasets are often available in two different return types: first return and bare earth. First return data include 

vegetation and are noisier, while bare earth data reflect the ground elevation underneath vegetation. The 

focus of this method is the extraction of beach slope along the section of beach between the dune toe 

and shoreline, which has little to no vegetation and will not be affected significantly when either first 

return or bare earth data are used. When available, bare earth data are preferred over first return data 

because of the extra quality control and editing that occurs with bare earth data processing.

Ground-based measurements of beach profiles may also be used to estimate beach slope, provided the profile covers the area of the beach between the dune toe and the shoreline. The same data processing techniques that are used to derive beach morphology from lidar data can be applied to groundbased beach profiles. 

Data Processing

Three-dimensional lidar data from each individual survey are gridded using a fixed-scale interpolator (Plant and others, 2002), which allows for variability in cross-shore and alongshore resolution, 

here, 2.5 m and 10 m, respectively. In addition to a gridded topographic surface, this method produces a 
2

corresponding grid of the residual error, which provides a measure of noise in the data. A Hanning filter 

with a width equal to two times the grid resolution was chosen to minimize noise in the data associated 

with vegetation, alongshore variability, and other error sources while preserving distinct morphologic 

characteristics. Analysis of cross-shore profiles of gridded data allows for automated extraction of morphologic features including the dune crest (xc

, zc), dune toe (xt, zt), shoreline position (xsl, zsl), and mean 

beach slope (βm) at 10-m alongshore intervals (fig. 1). Here, the shoreline is referenced to a mean-highwater (MHW) level position. Detailed descriptions of the algorithm used to extract shoreline position, 

dune crest, and dune toe can be found in Stockdon and others (2012). The mean beach slope is calculated using an endpoint method with the dune toe and the MHW shoreline as the landward and seaward 

limits, respectively. If a dune is not present, then the berm crest (xc,zc) becomes the landward limit. The 

averaged beach slope is computed every 200 m by applying a 400-m wide Hanning window and averaging over all survey times (fig. 2).

0 20 40 60 80 100 120 140 160 180

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Cross−shore distance (m)

Elevation (m, NAVD 88)

HWL

MHW

Profile

 x c

 , z c

x

t

 , z t

x

sl , z sl

m

Figure 1. Cross-shore profile of lidar-based elevations from the Outer Banks of North Carolina, surveyed in November 2012, indicating the 

locations and elevations of the dune crest (xc, zc), dune toe (xt, zt), shoreline (xsl,zsl), mean beach slope (βm), mean-high-water (MHW), and high 

water line (HWL). [m, meter; NAVD 88, North American Vertical Datum of 1988]. 

Uncertainty Analysis

Beach slope can vary with tidal cycles (hours to days), seasonal cycles (months), following 

storm events, or human-induced alterations to the coast, such as beach nourishment or scraping. Slopes 

also vary alongshore as the beach becomes narrower or wider. Because wave setup and swash elevations 

are sensitive to beach slope, estimating total uncertainty in the mean beach slope using all available 

survey information is necessary to compute the corresponding uncertainty in total water level predictions in advance of storm events. The total uncertainty in beach slope is defined as the mean residual 
3

0 4.5 9 18 KILOMETERS

0 4 8 2 MILES

North

Carolina

Duck

Nags Head

Oregon Inlet

Rodanthe

Salvo

Pamlico

Sound

75°45' W 75°30' W

36°15' N

36°00' N

35°45' N

35°30' N

Esri, HERE, DeLorme, MapmyIndia, © OpenStreetMap 

contributors, and the GIS user community

Average Beach Slope 

(radians)

EXPLANATION

0 - 0.05

0.05 - 0.06

0.06 - 0.07

0.07 - 0.08

0.08 - 0.09

0.09 - 0.10

Figure 2. An example of spatially and temporally averaged beach slopes computed using data collected 

between 1996 and 2012 for the Outer Banks of North Carolina.
4

error (MRE) of the raw slopes about the spatially and temporally averaged beach slope (see previous 

section). The mean residual error is defined as

(1)

where N is the number of raw beach slopes in each section, β is a single raw beach slope, and μβ is the 

spatially and temporally averaged beach slope. A 95 percent confidence interval is also provided, along 

with the MRE, and is defined as 2 × MRE (fig. 3). If more than 50 percent of data are missing in the 

400-m Hanning window, then the average beach slope and total uncertainty will not be computed. In 

general, the temporal variability in beach slope is an order of magnitude larger than the spatial 

uncertainty that exists within each 400-m Hanning window. 

Figure 3. Spatially and temporally averaged beach slopes for a 1-kilometer (km) section of coast in the Outer Banks of North Carolina. Each 

average slope (red circle) represents the spatial and temporal average of the raw points (small black circles) within ± 200 meters of the center 

point. The mean residual error (dark gray band) and 95 percent confidence interval (light gray band) are also shown.

9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Distance alongshore (km)

Beach slope (radians)

Raw beach slopes from all 13 surveys

95 % confidence interval

Spatially and Temporally Averaged Beach Slope

Mean residual error

EXPLANATION
5

References Cited

Plant, N.G., Holland, K.T., and Puleo, J.A., 2002, Analysis of the scale of errors in nearshore 

bathymetric data: Marine Geology, v. 191, no. 1–2, p. 71–86.

Sallenger, A.H., Krabill, W., Swift, R., Brock, J., List, J., Hansen, M., Holman, R.A., Manizade, S., 

Sontag, J., Meredith, A., Morgan, K., and Stockdon, H., 2003, Evaluation of airborne scanning lidar 

for coastal change applications: Journal of Coastal Research, v. 19, p. 125–133.

Stockdon, H.F., Doran, K.J., Thompson, D.M., Sopkin, K.L., Plant, N.G., and Sallenger, A.H., 2012, 

National assessment of hurricane-induced coastal erosion hazards—Gulf of Mexico: U.S. Geological 

Survey Open-File Report 2012–1084, 51 p., accessed March 10, 2015, at http://pubs.usgs.gov/

of/2012/1084/.

Stockdon, H.F., Holman, R.A., Howd, P.A., and Sallenger, A.H., 2006, Empirical parameterization of 

setup, swash, and runup: Coastal Engineering, v. 53, no. 7, p. 573–588.
Doran and others—A Method for Determining Average Beach Slope and Beach Slope Variability for U.S. Sandy Coastlines

—Open-File Report 2015-1053

ISSN 2331-1258 (online)

http://dx.doi.org/10.3133/ofr20151053