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Elevation errors in a LiDAR digital elevation model of West Seattle and their effects on slope

stability calculations

William C. Haneberg

Haneberg Geoscience, 10208 39th Avenue SW, Seattle WA 98146, USA, bill@haneberg.com

ABSTRACT

Comparison of 1719 differential GPS measurements with a 1 m LiDAR DEM covering

West Seattle shows that DEM elevation errors range from –4.88 m to +3.32 m. The errors are

spatially correlated with a semivariogram range of 125 m, an unclustered mean of –0.11 m, and

an unclustered standard deviation of ±0.75 m. Although there are statistically significant

correlations between elevation error and elevation, slope angle, and topographic roughness, the

relationships are weak and have little explanatory power. Monte Carlo simulations of slope

angle, static factor of safety, Newmark yield acceleration, and log Newmark displacement show

that elevation errors of the magnitude reported can have significant effects on derivative

calculations. The standard deviation of simulated slope angles increases from approximately ±2°

to ±3° as the true slope angle approaches zero. Errors in slope angles calculated from the LiDAR

DEM are smaller than those previously reported for a conventional 10 m DEM covering the

same area, but the decrease was not proportional to the decrease in DEM grid spacing. The

influence of elevation errors on static factor of safety errors is strongly dependent upon the slope

angle and decreases significantly as the slope angle increases. Effects on Newmark yield

acceleration and log Newmark displacement are not as profound, but are still large enough to

impart significant uncertainty into calculated results. Therefore, slope angle errors should be

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considered to be as significant as geotechnical parameter and pore water pressure uncertainties

when performing slope stability calculations based on high resolution LiDAR DEMs.

INTRODUCTION

High-resolution digital elevation models (DEMs) produced using airborne laser scanners

(commonly known as ALS or LiDAR) have become an increasingly important tool for landslide

hazard assessment in the Seattle area (Schulz, 2004; Coe and others, this volume) and elsewhere

(Falls and others, 2004; McKean and Roering, 2004; Derron and others, 2005; Haneberg and

others, 2005). In most cases, LiDAR elevation data are used to create shaded relief images that

are interpreted like aerial photographs and elevation errors are likely to be inconsequential. In

other cases, however, LiDAR data are used for quantitative mapping of attributes such as slope

angle, topographic roughness, topographic curvature, and physics-based models of shallow

landslide potential (Dietrich and others, 2001; McKean and Roering, 2004; Haneberg and others,

2005) and elevation errors have the potential to produce erroneous results.

Haneberg (2006), Holmes and others (2000), and Fisher (1998) have shown that the

effects of elevation errors on slope angles and other derivative values calculated from

conventional 10 m and 30 m DEMs can impart significant uncertainty into slope angle

calculations and values dependent upon them. McKean and Roering (2003), Hodgson and

Bresnahan (2004), and Adams and Chandler (2002) have evaluated LiDAR elevation errors

using root-mean-square (RMS) statistics, but the geostatistical variability of high resolution

LiDAR DEM elevation errors and their effects on geomorphic derivatives has not yet been

examined.

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This paper describes 1) the use of more than 1700 centimeter-accurate GPS

measurements to estimate errors in a 1 m horizontal resolution LiDAR DEM covering a portion

of Seattle, Washington and 2) an evaluation of the effects of elevation errors on infinite slope

stability calculations representative of conditions in the Seattle area. The study area is essentially

the same as that used by Haneberg (2006) for an evaluation of conventional 10 m DEM errors

(and, in fact, uses a superset of the GPS measurements from that study), which also allows the

two sets of results to be compared.

DATA COLLECTION

Study Area

The study area for this project is an 18 km2 portion of Seattle, Washington known as

West Seattle (Figure 1) and covered by the U.S. Geological Survey Seattle SW 7.5’ quadrangle

(formerly known as the Duwamish Head quadrangle). Relief in the area is approximately 155

meters, most of which occurs along coastal bluffs and a few deeply incised valleys. The study

area consists of glacially striated uplands underlain primarily by till and outwash of the

Pleistocene Vashon glacial stage (Troost and others, 2005), in some places modified by

landslides. Older glacial deposits and some bedrock are exposed locally, generally near the

bottom of coastal bluffs. Land use is primarily urban residential, with some clusters of low-rise

commercial buildings, schools, and parks (both open and heavily forested).

Global Positioning System Measurements

A total of 1723 differential elevation and horizontal position data were collected using

two Ashtech ProMark 2 single-frequency (L1) GPS receivers. The manufacturer’s specifications

for ProMark 2 survey accuracy are 0.005 m + 1 ppm baseline length (horizontal) and 0.010 m +

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2 ppm baseline length (vertical) for static surveys (Thales Navigation, 2002). The baseline length

refers to the distance between the GPS measurement location and a known reference station. For

stop-and-go kinematic surveys, the accuracies are 0.012 m + 2.5 ppm baseline length (horizontal)

and 0.015 m + 2.5 ppm baseline length (vertical). Repeated observations of benchmarks and

other known points suggest that both the accuracy and precision of the GPS data are in many

cases on the order of millimeters and rarely more than a centimeter or two. The general accuracy

of survey point locations was also checked by superimposing them on a geo-referenced 0.5 m

resolution orthophoto after each day of data collection.

Both static and stop-and-go kinematic survey methods were used. The static data were

recorded over intervals ranging from 18 minutes to 1 hour, depending on baseline length and

positional dilution of precision (PDOP) at the time. About two-thirds of the static data were

collected using one of the GPS receivers as a rover and the other as a base station occupying a

known control point established at a secure location within the study area, which produced

baseline distances of 1 to 10 km. The control point was established using repeated observations

of an hour or more with one of the GPS receivers and data from the nearby continuously

operating reference stations (CORS) RPT1 and SEAW. The remaining one-third of the static

points were established using one of the GPS receivers with the RPT1 and SEAW CORS as

control points, producing baselines of 10 to 20 km. These static points were then used as

initialization points for stop-and-go kinematic data collection along a number of traverses using

the second GPS receiver, which allowed for the rapid collection of many closely spaced data

necessary to evaluate the effects of DEM errors on slope angle maps (Haneberg, 2006). All of

the GPS data were processed using the instrument manufacturer’s Ashtech Solutions software,

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using NAD83 for the horizontal datum and NAVD88 for the vertical datum. Horizontal positions

were calculated in Washington state plane coordinates and orthometric heights were calculated

using the GEOID03 model incorporated into the Ashtech Solutions software.

One disadvantage of working in an urban area is that it is difficult to generate a network

of randomly located GPS measurement points or to measure elevations at locations very close to

specific LiDAR strike points as did Holmes and others (2000), Hodgson and Bresnahan (2004),

and McKean and Roering (2003), who conducted their studies in undeveloped areas. Data for

this study were necessarily collected in publicly accessible areas such as streets, rights-of-ways,

parks, and undeveloped parcels of land. Statistical declustering was used to help eliminate the

effects of non-random data locations when estimating global statistics. Another potential

problem with GPS surveys in many urban areas is interference with overhead utility lines.

Insofar as possible, the GPS measurements points for this study were also chosen so as to avoid

overhead utility lines. Successful collection of the stop-and-go kinematic data required that both

the rover and base units maintained a continuous lock on at least five satellites, which also made

it impossible to use that method beneath trees. Locations of the static and kinematic data points

are shown in Figure 2. Care was taken to collect many kinematic elevation measurements, which

were originally obtained to characterize errors in a conventional 10 m DEM (Haneberg, 2006), at

intervals as small as 10 m in order to characterize error variability over short distances.

Digital Elevation Model Creation

The DEM used for this project was produced from x,y,z bare-earth points supplied by the

Puget Sound LiDAR Consortium (PSLC). The LiDAR data were originally obtained by a private

contractor using a multiple-return laser scanner mounted in a fixed-wing aircraft, with an on-the-

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ground laser spot of approximately 0.9 m, pulse spacing of 1.5 m, and 50% overlap between

adjacent flight lines. This produced coverage with an average density of approximately 1

pulse/m2 and a stated vertical accuracy of approximately 0.30 m (with the caveat that the

accuracy is likely to be less in some areas). As such, the dataset used in this study is typical of

the commercial LiDAR data that is becoming increasingly available for landslide hazard

assessment projects.

The x,y,z points were supplied with units of U.S. survey feet in the Washington state

plane coordinate system and NAD 83 datum (1991 HPGN adjustment). They were imported into

a 1 m (3.2808 feet) raster grid that was filled using inverse distance squared interpolation. In

cases where more than one x, y, z point fell into a raster, only the last value was retained. An

alternative, which was not evaluated for this project, would have been to calculate a mean or

median value for each raster with multiple values. Although some references recommend against

the use of inverse distance interpolation for terrain modeling (e.g., Maune and others, 2001),

which can produce unrealistic dimpled surfaces from sparse data, practical experience has shown

the method to be a computationally fast and viable alternative for dense LiDAR data sets as long

as care is taken to avoid interpolation artifacts (e.g., Haneberg and others, 2005). Unlike the

specialized topographic interpolation algorithms incorporated into some proprietary GIS

software, inverse distance algorithms are freely available for use on many computer systems.

Horizontal coordinates were converted from U.S. survey feet to meters by recalculating

coordinates for the four corners of the DEM and changing the raster size units from feet to

meters. Neither the datum nor the coordinate system was changed by this procedure. Heights

were converted from feet to meters by simple multiplication. As received from PSLC, the

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LiDAR data included several clusters of elevation values for the waters of Puget Sound.

Therefore, the final step of DEM creation was to remove elevations less than 2.5 m in order to

eliminate most of the spurious values. This also had the effect of removing 4 of the 1723 GPS

elevation measurements. The final DEM thus contained 19,871,336 elevation points for

comparison against 1719 GPS measurements.

ELEVATION ERRORS

Method of Calculation

Elevation errors were calculated for each 1 m raster containing a GPS measurement by

subtracting the DEM value for that raster from the GPS value, and the entire difference was

attributed to DEM elevation error (cf. Hodgson and Bresnahan, 2004). This procedure lumps

together true LiDAR vertical errors, true LiDAR horizontal errors, DEM interpolation errors, and

GPS surveying errors. This study assumes that GPS surveying errors are on the order of a

centimeter or two, or about the same order of magnitude as interpolation errors, based upon both

the manufacturer’s instrument specifications and field testing. As described below, the errors

calculated during this project were compared with geomorphic factors such as slope angle and

topographic roughness but no attempt was made to subdivide errors on the basis of ground cover

or land use.

Global Statistics

The calculated elevation errors ranged from –4.88 m to +3.32 m, with the majority falling

in the range of ±0.5 m. Both clustered and unclustered summary statistics were calculated for the

elevation errors in order to account for the non-random sampling pattern. The clustered error

data had a mean of –0.03 m and a standard deviation of ±0.65 m, and cell declustering (Isaaks

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and Srivastava, 1989) produced an unclustered mean of –0.11 m and an unclustered standard

deviation of ±0.75 m (Figure 3).

There are no strong relationships between the magnitude of elevation error and elevation,

maximum slope angle, slope aspect, or topographic roughness (Figure 4). The maximum slope

angle for each raster was defined as the largest of the four finite difference slope values

calculated from the eight surrounding DEM values. Topographic roughness was calculated using

the method described by Haneberg and others (2005), in which roughness is defined as the

standard deviation of residual topography within a moving window of specified size (in this case

5 by 5 rasters). Residual topography, in turn, was defined as the difference between the 1 m

DEM elevations and a topographic map smoothed with a 5 by 5 raster averaging window. Linear

regression showed that there are statistically significant (p ! 0.05) relationships between the

absolute value of elevation error and elevation, slope angle, and topographic roughness but not

between elevation error and slope aspect (Table 1). The goodness of fit was low to virtually

nonexistent in all three cases for which a statistically significant relationship exists, however, and

these relationships have little explanatory power. Taken together, the three variables account for

only 30% of the variability of the elevation error.

Spatial Correlation Structure

The structure of spatially correlated data is conventionally represented using the

semivariance, which is the square of the difference between pairs of points separated by a given

distance known as the lag. The semivariance is conventionally written as (e.g., Isaaks and

Srivastava, 1989; Goovaerts, 1997)

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!

" h( ) =z x

i+ h( ) # z xi( )[ ]

2

N h( )i=1

N (h )

$ (1)

in which ! (h) is the semivariance, h is the lag, z is the dependent variable being measured, x is

the location of the measurement, and N(h) is the number of pairs separated by h.

To calculate an empirical semivariogram, pairs of points with similar lags are binned

together as in a histogram and the semivariance of all pairs of points within each bin is

represented by a single point. This is done because it is unlikely real data will contain enough

pairs with identical lags to produce reliable estimates of the semivariance. Theoretical or

modeled semivariograms, in contrast, consist of mathematical functions. The gstat package for

the statistics software R was used to produce the semivariograms in this paper (Pebesma, 2004;

R Development Core Team, 2006).

The geometry of semivariograms is described in terms of nuggets, sills, and ranges as

illustrated in Figure 5. The nugget is the semivariance at a lag of zero. When present, the nugget

represents the variability of multiple measurements at the same location, for example as a

consequence of instrumental error. Absent measurement errors, there should be no difference in

measurements made at the same location. The sill is the plateau reached by the semivariance at a

lag known as the range. In general, the sill is expected to equal the variance of the data.

An omnidirectional empirical semivariogram with a bin width of 10 m was calculated

using the Cressie and Hawkins (1980) robust method to allow for non-normality of the error data

and the effects of outliers. The empirical semivariogram (Figure 6) shows a well-developed

spatial correlation structure over lag distances up to several kilometers. Oscillations in the

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empirical semivariogram, sometimes referred to as hole effects, reflect periodicity in the spatial

variability of errors that are likely related to the striated glacial topography over lags of several

kilometers. Erratic empirical semivariogram behavior at large lags (> 6000 m) may be the result

of an underlying very large scale correlation structure and the decreasing numbers of widely-

spaced pairs available for semivariogram calculation.

Over the short lags relevant to this study (Figure 6B), the empirical semivariogram can be

modeled using a theoretical exponential semivariogram with a range of 40 m and a sill of 0.20,

which has the form

!

" h( ) = 0.20 1# exp #h /40 m( )[ ] (2)

The theoretical semivariogram was determined by plotting a trial curve, visually assessing the fit

(particularly for lags < 100 m), and adjusting the sill and range values until a reasonably good

match was found. In this case the sill is appreciably less than the variance of the data (0.20 vs.

0.752 = 0.56) because the theoretical semivariogram was determined using only pairs with lags !

3500 m.

EFFECTS ON SLOPE STABILITY CALCULATIONS

The effect of DEM elevation errors on slope stability calculations is tempered by the facts

that the errors seem to be correlated over distances ranging from meters to kilometers (Haneberg,

2006; Holmes and others, 2000; Fisher, 1998) and that slope angles are calculated from

neighboring elevation values separated from each other by distances of

!

2"x , where !x is the

DEM grid spacing. The relevant measure of elevation error or uncertainty in this study is

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therefore the variance or standard deviation of DEM elevation errors among grid points located

approximately 2 meters apart, not the global variance or standard deviation.

Equation (1) can be written as (Isaaks and Srivastava, 1989; Goovaerts, 1997)

!

" h( ) = 1

2E # x( ) $# x + h( )[ ]

2{ } =Var # x( ) $# x + h( ){ } (3)

in which " is the elevation error, E is the expectation operator, and Var is the variance operator.

For typical slope angle calculations,

!

h = 2"x or

!

2 2"x , depending whether the slope angle is

calculated parallel or obliquely to the DEM grid. Equation (3) can be rearranged to give the

variance of elevation errors among points located

!

2"x from each other, which is

!

Var " x( ) #" x + 2$x( ){ } = % 2$x( ) (4)

For the model semivariogram given by equation (2), the elevation error variance appropriate for

geomorphic derivative calculations on a 1 m grid (h = 2 m) is 0.0098 m2. Its square root, the

elevation error standard deviation, is thus approximately ±0.10 m (compared to global clustered

and unclustered standard deviations of ±0.65 m and ±0.75 m, respectively).

The effect of DEM elevation errors on slope stability calculations was evaluated using a

series of Monte Carlo simulations using the method described in Haneberg (2004 a). In each of

the simulations, elevation errors were randomly selected from a normal distribution with a mean

of zero and a standard deviation of ±0.10 m, consistent with the values measured in this study,

and then added to grids of elevation values defining planes with a slope angles of 10°, 20°, 30°,

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and 40°. A total of 1000 random elevation realizations were generated for each of the four

slopes. Slope angles were calculated for each realization using the Zevenbergen and Thorne

(1987) equations, and the process was repeated to produce an ensemble of slope angle

calculations for each of the four slopes. Each slope angle realization was used to calculate a

static factor of safety against sliding using the infinite slope approximation

!

FS =cT

+ "MT cos

2 # tan$

"MT sin# cos#

(5)

in which cT is the total cohesion consisting of both soil cohesion and root strength (Pa), !M is the

moist soil unit weight (N/m3), # is the angle of internal friction, T is the soil thickness (m), and $

is the slope angle (degrees). In order to evaluate only the effects of DEM elevation errors, all of

the variables except slope angle were held constant and the effects of pore water pressure were

not considered. The following representative values were chosen based upon a tabulation in

McCalpin (1997) and a requirement of static stability throughout the study area: cT = 10 kPa, !M

= 19 kN/m3, T = 2 m, and # = 35°. The static factor of safety was then used to calculate a

dimensionless Newmark yield acceleration

!

aN

g= (FS "1)sin# (6)

in which g is gravitational acceleration, for each realization. Newmark displacement was

estimated using the simplified empirical relationship developed by Jibson and others (2000):

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!

logDN =1.521 log IA "1.993 logaN

g"1.546 (7)

in which DN is the displacement (cm) as the result of seismic shaking and IA is the observed or

predicted Arias intensity (m/s). Equation (7), which has a strong but not perfect goodness-of-fit

of r2 = 0.83, adds some additional uncertainties were not considered in this study. The results

shown in this paper were calculated by assuming that IA = 4 m/s.

Results the Monte Carlo simulations are shown as a series of histograms in Figure 7. The

standard deviation of the simulated slope angles decreases as the average slope angle increases,

and the simulated slope angles follow nearly symmetric distributions. Because the DEM points

are separated by only 1 m, even small elevation errors can give rise to slope angle errors of

several degrees. The calculated factor of safety distributions are markedly asymmetric and

decrease in width as slope angle increases, meaning that static factors of safety calculated for

gentle slopes will generally less reliable than values calculated for steep slopes. The geotechnical

parameter used were deliberately chosen to ensure that FS > 1 for dry static conditions, which is

reflected in the Monte Carlo simulation results. The simulated Newmark yield acceleration

distributions change from nearly symmetric to asymmetric while becoming narrower as the mean

slope angle increases, with standard deviations ranging from ±0.066 g for the 10° average slope

to ±0.039 g for the 40° average slope. Unlike the other simulated results, the logarithmic (base e)

displacement uncertainty increases with average slope angle.

The relationship between average slope angle and simulated slope angle errors can be

understood using the first-order, second-moment analytical approximation for slope angle error

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developed by Haneberg (2004 b) for finite difference slope angle calculations typically used on

DEMs. Rewritten in terms of the slope angle standard deviation, Haneberg’s (2004 b) analytical

approximation is:

!

s" =180

#

8 s$%x

4%x 2 + zr,c+1 & zr,c&1( )

2

+ zr+1,c & zr&1,c( )

2 (8)

in which

!

s" is the standard deviation of the slope angle in degrees,

!

s" is the standard deviation of

the elevation errors for the four points used in the slope angle calculation, !x is the DEM grid

spacing, z is elevation, and r and c are the DEM row and column indices. Equation (8) assumes

that the elevation error standard deviation is the same for all four elevation values used in the

calculation. The presence of elevation differences in the denominator of equation (8) requires

that the elevation error decrease as the slope angle increases. Figure 8 shows the relationship

defined in equation (8) as a function of summed squared elevation differences,

!

zr,c+1 " zr,c"1( )

2

+ zr+1,c " zr"1,c( )

2

, for slope angles calculated from a 1 m DEM grid with an

elevation error standard deviation of ±0.10 m. As shown in Figure 8, the Monte Carlo

simulations and the analytical approximation are in general agreement, although the two diverge

slightly at low elevation differences (i.e., small slope angles). The divergence occurs because the

analytical approximation assumes normally distributed slope angles. The Monte Carlo

simulations, in contrast, cannot not produce negative slope angles and the results therefore

become increasingly asymmetric and non-normal as the elevation difference approaches zero.

DISCUSSION

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Centimeter-accurate differential GPS surveying has shown that elevation errors in a 1 m

LiDAR DEM covering a portion of Seattle have an unclustered mean of –0.11 m and an

unclustered standard deviation of ±0.75 m, compared to a stated accuracy of approximately

±0.30 m. The standard deviation of ±0.75 m is two to three times the root mean square errors

reported by Hodgson and Bresnahan (2004), Adams and Chandler (2002), and McKean and

Roering (2003), and less than the ±0.84 m standard deviation reported by Norheim and others

(2002) in their assessment of lidar DEM errors in an area northeast of Seattle. By way of

comparison, it is also considerably smaller than the standard deviation of ±2.36 m reported by

Haneberg (2006) from his study of elevation errors in a conventional 10 m DEM covering the

same area and the values reported by Fisher (1998) and Holmes and others (2000) in their

analyses of 10 m and 30 m DEMs from other areas.

The measured elevation errors are positively but weakly correlated with slope angle.

topographic roughness, and, to a lesser degree, elevation. In all three cases r2 values are low and

the relationships have little explanatory power, together accounting for no more than 30% of the

elevation error variability. To a first approximation, therefore, the elevation errors measured in

this study were considered to be largely independent of elevation, slope angle, slope aspect, and

topographic roughness.

Another difference between the LiDAR DEM elevation errors described in this paper and

conventional DEM elevation errors described elsewhere is the spatial correlation structure. The

errors described in this paper were modeled by a single component semivariogram with a range

of 40 m, whereas the conventional DEMs analyzed by Haneberg (2006), Fisher (1998), and

Holmes and others (2000) required multi-component semivariograms with maximum ranges on

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the order of a kilometer. The shorter semivariogram range for the LiDAR means that, although

the global mean and variance can be estimated using a relatively small number of randomly

located measurements, elevation measurements must be closely spaced in order to obtain large

numbers of pairs separated by distances of 102 m or less, which is the order of magnitude of the

semivariogram range obtained in this study (in this study, data were separated by distances as

small as 10 m). If elevation measurements are clustered, as they necessarily were in this study,

then the reliability of elevation error estimates across the study area may be variable if the areas

in which data were collected are not representative of the entire area of concern.

This paper does not consider three factors that may influence the effects of DEM errors

on slope stability calculations. First, differences in errors associated with different interpolation

methods were not considered. Practical experience with a variety of LiDAR data sets suggests

that differences arising from the choice of interpolation method are likely to be most significant

in areas of widely separated LiDAR ground strikes. For example, some public domain LiDAR

data sets in the Puget Sound region contain areas with 30 m or more between ground strikes in

steep forested areas. In most cases, however the ground strike spacing in steep forested areas is

typically on the order of meters. Second, the weak correlations between elevation error, slope

angle, and topographic roughness were not incorporated into the Monte Carlo simulations but

may be important in more detailed studies. Third, the validity of the infinite slope approximation

in cases where the DEM grid spacing is of the same order (or less) of the typical soil thickness

was not investigated. The infinite slope approximation has proven to be useful and mechanically

justifiable when used with 10 m or 30 m DEMs, but its mechanical validity for 1 m or 2 m

DEMs remains unclear. One alternative might be to average LiDAR slope angles over areas

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known to be appropriate for infinite slope approximations, say 10 m by 10 m moving windows,

in which case the significance of elevation errors may be different than that described in this

paper. All three of these issues deserve attention in future studies.

Elevation errors of the magnitude described in this study can impart significant errors

into slope angle and slope stability calculations based on high-resolution DEMs. Although the

LiDAR DEM represents a significant improvement over the conventional 10 m DEM covering

the study area, it is not error free. LiDAR derived slope angles should be expected to have

standard deviations on the order of ±2° to ±3°. Knowledge of slope angle errors is particularly

important if slope angle, static factor of safety, or Newmark displacement maps are used as input

for GIS-based models in a regulatory or public safety setting, because slope angle standard

deviations of several degrees can have a significant effect on output. Haneberg (2006) discusses

how the effects of slope angle errors can be conservatively incorporated into hazard zonation

maps based on factors of safety or Newmark displacements by specifying a maximum acceptable

probability of misclassification.

The geotechnical parameters used in this study were deliberately chosen to ensure that FS

" 1 for dry slopes under static conditions. Slightly lower cohesive strength or angles of internal

friction, or non-zero pore water pressures, could lower the mean factor of safety enough that

some slopes would have a mean FS > 1 but a non-trivial probability of sliding (FS < 1). Results

such as those described in this paper may help to establish minimum acceptable mean factors of

safety for zoning or hazard delineation by taking into account the imprecision introduced by

slope angle errors and geotechnical parameter uncertainties (see Haneberg, 2000, 2004 b for

more detailed discussion).

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The magnitude of slope angle errors arising from elevation errors in the high-resolution

LiDAR DEM used in this study are only slightly smaller than those calculated for conventional

10 m and 30 m DEMs by Haneberg (2006), Holmes and others (2000), and Fisher (1998).

Although a high resolution LiDAR DEM is capable of representing smaller scale topographic

features than a more coarsely gridded conventional DEM, and thus reduces the component of

slope angle error introduced by ignoring sub-raster topography, the slope angle values calculated

from neighboring points will likely have the same magnitude of uncertainty. This result can be

understood by considering the Haneberg (2004 b) analytical approximation for slope angle

uncertainty given in equation (7). For the baseline case of zero slope, the slope angle standard

deviation given by equation (7) becomes

!

s" =1

2

180

#

s$

%x (8)

Equation (8) shows that the decrease in elevation error must be proportional to the decrease in

DEM grid spacing if the slope angle variance is to remain constant. Inclusion of the elevation

values will introduce a dependence on slope angle but the fundamental reciprocal relationship

between elevation error and DEM grid spacing remains intact. Equation (8) predicts a maximum

slope angle standard deviation of 4.7° for the conventional 10 m DEM elevation error data in

Haneberg (2006) and 3.3° for the LiDAR data in this paper. Thus, reducing the DEM grid

spacing by a factor of 10 reduced the slope angle standard deviation by a factor of only 1.4 in

this study area.

ACKNOWLEDGEMENTS

This research was supported by U.S. Geological Survey NEHRP FY 2004 award

04HQGR0035. The LiDAR elevation data were provided by the Puget Sound LiDAR

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Consortium. Use of specific product names is for informational purposes only and does not imply

endorsement by the U.S. Geological Survey or the author. Comments by reviewers Kevin Schmidt

and David Tarboton improved the quality of this paper and are greatly appreciated.

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TABLE 1. ELEVATION ERROR LINEAR REGRESSION RESULTS FOR FOUR DIFFERENT INDEPENDENT VARIABLES

Variable Intercept Slope r2 p

Elevation 0.33 0.0010 0.0057 0.001

Slope angle 0.10 0.024 0.19 < 2.2 x 10-16

Slope aspect 0.46 -0.00021 0.0014 0.06

Roughness 0.18 3.00 0.11 < 2.2 x 10-16

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FIGURE CAPTIONS

Figure 1. Index map showing the general location of the study area relative to the Puget Sound

region.

Figure 2. Locations of the static and kinematic GPS data points used in this study superimposed

on a shaded relief image of the 1 m LiDAR digital elevation model (DEM) of West Seattle.

Coordinates are in the Washington State Place Coordinate System (North) using the NAD83

(HPGN) horizontal datum, with1000 m ticks shown in the margins.

Figure 3. Histograms of clustered (top) and cell-declustered (bottom) elevation errors measured

in this study, normalized so that the area under the bars of each histogram sums to unity.

Figure 4. Empirical relationships between the magnitude of the measured elevation error and

elevation, slope angle, slope aspect, and topographic roughness.

Figure 5. Idealized empirical and theoretical semivariograms illustrating the geometric meaning

of the nugget, range, and sill.

Figure 6. Variograms showing the spatial correlation of elevation errors in the study area. The

open circles are empirical values calculated for bin widths of 10 m. The solid line is the

theoretical semivariogram described by equation (2), with a sill of 0.20 m2 and a range of 40 m.

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A) Empirical and theoretical semivariograms for all pairs with lag ! 10,000 m. B) Close-up

showing all pairs with lag ! 3500 m.

Figure 7. Results of Monte Carlo simulation of the effect of observed elevation errors on slope

angles and slope aspects calculated for hypothetical north-facing slopes with true slope angles of

10°, 20°, 30°, and 40° and an elevation error standard deviation of ±0.10 m for points 2 m apart.

Each row consists of one simulation result for a given true slope angle and each of the four

simulations consists of 1000 realizations of slope angle, static factor of safety, Newmark yield

acceleration, and log Newmark displacment.

Figure 8. Calculated slope angle standard deviations as predicted by Monte Carlo simulations

from Figure 7 and the first-order, second-moment approximation of Haneberg (2004b).

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Haneberg Figure 1

27

Haneberg Figure 2

28

Haneberg Figure 3

29

Haneberg Figure 4

30

Haneberg Figure 5

31

Haneberg Figure 6

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Haneberg Figure 7

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Haneberg Figure 8