Measurement of Slope Deformation 

Using Quadrilaterals

U.S. GEOLOGICAL SURVEY BULLETIN 1842-B
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Chapter B

Measurement of Slope Deformation 

Using Quadrilaterals

By Rex L. Baum, Arvid M. Johnson, and Robert W. Fleming

A description of a new method for measuring 

displacement at the boundaries of a landslide and 

strain and tilt at the surface of a landslide

U.S. Geological Survey Bulletin 1842

LANDSLIDE PROCESSES IN UTAH-OBSERVATION AND THEORY
DEPARTMENT OF THE INTERIOR 

DONALD PAUL MODEL, Secretary

U. S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director

UNITED STATES GOVERNMENT PRINTING OFFICE: 1988

For sale by the

Books and Open-File Reports Section

U.S. Geological Survey

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Box 25425

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Library of Congress Cataloging in Publication Data

Measurement of slope deformation using quadrilaterals.

(U.S. Geological Survey bulletin ; 1842) (Landslide processes in 

Utah observation and theory ; ch. B)

Bibliography: p.

1. Landslides Utah. 2. Slopes (Soil

mechanics) Utah. 3. Quadrilaterals. I. Johnson, Arvid M. II. Fleming, 

Robert W. III. Title. IV. Series. V. Series: Landslide processes in 

Utah observation and theory ; ch. B. 

QE75.B9 no.1842-B 557.3 s [551.3'53] 88-600370 [QE599.U]
CONTENTS

Abstract Bl 

Introduction Bl 

Acknowledgments B2 

Method B2

Computational methods B2

Field technique B5

Errors and sensitivity B6 

Examples of quadrilateral measurements B7

The Aspen Grove landslide B7

Measurement of displacement on a landslide flank B8

Measurement of displacement at a landslide toe BIO

Measurement of strain and tilt on a ridge B13

Measurement of strain and tilt in an area of developing normal faults within

the landslide B16 

Discussion and conclusion B17 

Appendix B21 

References cited B23

FIGURES

Bl. Sketch showing typical quadrilateral. B2

B2. Sketch of reference system for computing displacements across a

discontinuity or landslide boundary B4 

B3. Diagram showing geometrical relationships between a deformed

triangle and its stretch ellipse B5

B4. Diagram showing dip direction of triangle ABC B5 

B5. Map showing location of the Aspen Grove landslide, central

Utah B8

B6. Map showing the Aspen Grove landslide, August 1984 B9 

B7. Sketch of the site of Q-2 on June 15, 1983 BIO 

B8. Map showing part of the Aspen Grove landslide in vicinity of

quadrilateral Q-2 during August 1984 fill 

B9. Map and cross section showing internal toe of the Aspen Grove

landslide B12 

BIO. Graph showing displacements of stake A at Q-27 from May 18 to

August 4,1984 B13

Bll. Enlarged view of area "d", shown in figure B9 B14 

B12. Map showing ridge on the right flank of the Aspen Grove landslide,

August 1984 B15 

B13. Polar plot showing attitude (dip and dip directions) of the four

triangles in quadrilateral Q-34 B16 

B14-B16. Map showing part of the Aspen Grove landslide in the vicinity of Q-5

during:

B14. August 1984 B18

B15. August 1985 B19

B16. September 1986 B20

Contents
TABLES

Bl. Principal strains at Q-34 B16 

B2. Strains in Quadrilateral Q-5 B17

CONVERSION FACTORS

Multiply_______By_____To obtain 

mm (millimeters) 0.03937 in. (inches) 

cm (centimeters) 0.3937 in. (inches) 

m (meters) 3.2808 ft (feet) 

km (kilometers) 0.6214 mi (miles)

IV Contents
LIST OF SYMBOLS

a, b, c arbitrary constants

A, B, C, D corners of a quadrilateral

d area strain

L length of a line segment

p plunge angle of the displacement vector

S stretch

S,, S2 major and minor principal stretches

U magnitude of the displacement vector

u, horizontal displacement component parallel to the reference line and

	the x axis

uz horizontal displacement component perpendicular to u,

u3 vertical displacement component

w error of closure, an angle

x, y, z cartesian coordinates, z is vertical

a angle between S^ and a reference direction, f, in the deformed state

jS, angle between S, and a line in the deformed state

 y finite shear strain

5 dip angle of a plane

£ direction of a reference line (usually a diagonal of the quadrilateral)

	used in computing the direction of the major principal stretch 

0, angles between a side and a diagonal of the quadrilateral 

\ angle between the reference line and the horizontal component of

	displacement 

£, an arbitrary direction in the x-y plane (usually the trend of a side or

	a diagonal of a quadrilateral) 

4>, horizontal projections of the angles 0/ 

w, angle between £, and the positive x direction 

wmax dip direction of a plane tangent to the ground surface 

e linear strain, S-1

Primes denote positions or values in the deformed state

Contents

LANDSLIDE PROCESSES IN UTAH-OBSERVATION AND THEORY

Measurement of Slope Deformation 

Using Quadrilaterals

By Rex L Baum1 , Arvid M. Johnson2 , and Robert W. Fleming1

Abstract

Quadrilaterals, four-sided figures defined by survey 

stakes at their four corners, were used to measure 

deformation of the ground surface at a landslide in central 

Utah. The lengths of the sides and diagonals and the vertical 

distances between the corners of each quadrilateral were 

measured at intervals of days or weeks to observe changes.

Equations were derived for computing threedimensional displacements of corners of a quadrilateral that 

straddles a landslide boundary. Equations were also derived 

for computing the magnitude and orientation of principal 

stretches and the initial and final dips and dip directions of 

each of four triangles defined by the perimeter and diagonals 

of a quadrilateral on the surface of a landslide.

Length measurements were made with a steel tape and 

elevation differences between the tops of stakes were 

measured by taking backsights with a precision level. For the 

measurement technique used, we observed the following: 

Errors of 5-6 mm occurred in 10-20 percent of our length 

measurements. Because the landslide moved at rates of 

several millimeters to several centimeters per day, 

displacements could be computed from data sets measured 

at intervals of two or three days. However, strains could be 

computed only from data sets measured at intervals of many 

weeks. Square quadrilaterals, from 1.5 to 4.0 m wide, worked 

best for observing deformation.

Twenty-eight quadrilaterals were used to measure 

deformation on a long, narrow landslide; in general, the 

quadrilaterals functioned very well. For example, at a small 

quadrilateral on the flank of the landslide, displacements of a 

few meters occurred before the stakes had to be relocated in 

order to continue using the quadrilateral. Several centimeters 

of displacement occurred at a quadrilateral on an internal toe 

of the landslide before thrust faults appeared at the surface of 

the toe. Tilt of 1°-2° was measured at a quadrilateral on the

80225.

^.S. Geological Survey, Box 25046, M.S. 966, Denver, CO 

2Purdue University,West Lafayette, Indiana.

side of a deforming flank ridge. Another quadrilateral 

documented the deformation that preceded the appearance 

of a normal fault within the boundaries of the landslide.

INTRODUCTION

Displacements and strains have been used in many 

studies of slope behavior. Such data are useful for 

studying landslide kinematics and dynamics (Keefer and 

Johnson, 1983; Iverson, 1984; Lantz, 1984), monitoring 

stability of slopes and safety of structures during and 

after construction (Hanna, 1985), planning and 

evaluating landslide stabilization schemes (TerStepanian, 1984), measuring near-surface stresses in 

slopes (Zaruba and Mencl, 1982), and estimating the 

geometry of slip surfaces (Carter and Bentley, 1985; 

Cruden, 1986). These applications have in common an 

attempt to use surface measurements to infer conditions 

at depth.

In order to understand and possibly predict some 

aspects of landslide behavior by means of surface 

measurements, monitoring was begun at some of the 

landslides that occurred in Utah during 1983. 

Quadrilaterals were installed at several places on a 

landslide in central Utah, at the suggestion of David 

Varnes (U.S. Geological Survey). Such a technique was 

also proposed by Brunsden (1984, p. 393, 396), who 

indicated that no published accounts exist of the use of 

quadrilaterals in studying landslides.

In surveying, a quadrilateral is a four-sided, plane 

figure defined by the survey stations (stakes or 

monuments) at its four corners (Ingram, 1972; Brinker 

and Wolf, 1977; Laurila, 1983). Brinker and Wolf (1977, 

p. 372) noted that the quadrilateral is the simplest 

geometric figure that permits rigorous closure checks

Measurement of Slope Deformation B1
and adjustment of measurement errors. Errors of closure 

can be determined because a redundant measurement is 

made whenever the lengths of all four sides and both 

diagonals of the quadrilateral are measured. Various 

methods are used to distribute the error (Laurila, 1983).

In addition to the fact that the measurements of 

quadrilaterals can be checked and adjusted rigorously, 

quadrilaterals have several features that make them 

useful for measuring strains and displacements. 

Quadrilaterals permit measurement of strain in several 

directions at one location because the sides and diagonals 

trend in as many as six different compass directions. 

Because quadrilaterals are defined by survey stations at 

their four corners, soil within a quadrilateral can undergo 

large deformations without the quadrilateral affecting 

soil movement. This is a distinct advantage over some 

types of soil strain gages (Hanna, 1985, p. 296-300). 

Quadrilaterals can be measured with a variety of 

instruments, such as a steel tape, a tape extensiometer, or 

a sophisticated electronic device. Thus, they can be used 

to make measurements at levels of precision needed for 

a wide variety of landslide studies.

Experience with the use of quadrilaterals during 

the last few years has convinced us that the method has 

significant potential for studying landslide kinematics. 

We have learned, by trial and error, how quadrilaterals 

can be expected to function in field situations. This paper 

explains the theory behind the use of quadrilaterals and 

describes how quadrilaterals were used in the field. The 

use of the data for describing the kinematics of the 

landslide is discussed in another chapter of this bulletin.

optimum range of size, shape, and frequency of 

measurement of the quadrilaterals; and detecting and 

dealing with the measurement errors.

Computational Methods

In the field, a quadrilateral is marked by stakes or 

permanent markers at its four corners (fig. Bl). 

Measurements of the lengths of the sides and diagonals 

are repeated to observe deformation of the quadrilateral 

through time.

If the quadrilateral straddles an outer edge of the 

landslide, then the data are used to compute landslide 

displacements relative to the stable ground along the 

boundary. If the quadrilateral is within the landslide, then 

the data are used to compute strains and tilts within the 

landslide mass.

Errors. Detailed theories for checking closure and 

adjusting measurement errors in quadrilaterals are in the 

surveying literature (Ingram, 1972; Brinker and Wolf, 

1977; Laurila, 1983; and others).

We compute the error of closure by projecting the 

quadrilateral onto a horizontal plane. Angles 4> lt ..., <£ 8 , 

the horizontal projections of angles 0lv.., 08 , shown in 

figure 1, are computed by means of the law of cosines. 

The angular error of closure is determined by using the 

following equation (Laurila, 1983, p. 43):

w

« =(!>< )- 360° (1)

ACKNOWLEDGMENTS

David J. Varnes (USGS, Denver) suggested the 

use of quadrilaterals and helped with fieldwork. Roger 

Johnson (Ephraim, Utah), Chad Parry (Ephraim, Utah), 

Douglas Robins (Gunnison, Utah), Roger Nichols 

(USGS, Denver), Lee-Ann Chandler (USGS, Denver), 

and Gary Jorgenson (USFS, Ephraim, Utah) helped with 

fieldwork. Ben Black (Manti District Ranger, USFS, 

Ephraim, Utah) shared observations and helped solve 

various logistical problems. We express our sincere 

appreciation to all these people.

METHOD

Here we discuss the theory behind the use of 

quadrilaterals and some practical details that apply to all 

quadrilaterals: the fundamentals of computing 

displacements, strains, and tilts (derivations of the 

equations for finite strain are in Appendix); the

Displacement Displacement across a landslide 

boundary is computed from the original and final 

positions of a corner of the quadrilateral (fig. B2). Thex-y

Figure B1. Sketch showing typical quadrilateral. (A) 

Isometric View, (B) Normal view. Stakes are labeled A 

through D in clockwise rotation, with stake A having the 

highest elevation. Slope lengths of the four sides and two 

diagonals and the vertical distances (elevation differences) 

between A and the other three corners are measured. Angles 

9,,.... 98 are computed from the slope lengths of the sides 

and diagonals of the quadrilateral, by the law of cosines.

B2 Landslide Processes in Utah
plane is horizontal, with the positive x-axis being the 

horizontal projection of side AD of the quadrilateral 

(figs. BIB, B2A). Angle tf> (fig. B2B) for the initial 

position of corner B of the quadrilateral is computed 

from the horizontal components of AS, BD, and DA by 

means of the law of cosines. Similarly, angle 0' for the 

final position of corner B, is computed from the 

horizontal components of AS", B'D, and DA. The x andy 

components of line AB and the x and y components of 

line AB' are computed by means of standard 

trigonometric formulas. The x and y components of 

displacement, u± and u2 , (fig. B2) are determined by:

and,

u i =x '-x ,

u z =y'-y.

(2a) 

(2b)

The vertical component of displacement, w3 , is the 

elevation difference between B and B'. Total 

displacement (net slip), U, is determined by the 

Pythagorean theorem:

JT U -

2\l/2 (3)

The counterclockwise angle \ between the positive 

x-axis and the horizontal component of displacement of 

corner B is:

\ =tan- 1 (w 2/Ml ). (4) 

Plunge,/?, of the displacement is determined by:

p = (5)

In this way, one determines the magnitude, 

azimuth, and plunge of displacements at a point on the 

landslide. Similar calculations are performed to compute 

displacements of the second point, C.

Strains. The differences in displacement of 

corners B and C of the quadrilateral, used to measure 

displacement at a landslide boundary (fig. B2A), define 

the strain between the corners. Also, by comparing the 

original length of line EC to its final length, B 'C', stretch 

parallel to B'C' can be determined.

More detailed information about the strain can be 

obtained at a quadrilateral on the surface of the 

landslide. If all four corners of the quadrilateral are 

coplanar and remain so during deformation, then one set 

of principal strains for the quadrilateral can be computed 

by the method of least squares (Appendix). However, the 

corners seldom are coplanar. Therefore, a set of principal 

strains is computed, in the plane of the triangle, for each 

of the four triangles making the quadrilateral (fig. Bl), 

AABC, ABCD, ACDA, ADAS. Each set of strains

comprises the major and minor principal strains and the 

angle between the major principal strain and a reference 

direction for the quadrilateral.

Strains larger than 0.01 (1.0 percent) are treated as 

finite strains. A convenient measure of finite strain is the 

stretch1 , S (Malvern, 1969; A.M. Johnson, unpub. data, 

1981), which is the ratio of the final length, Lf) to the 

initial length, L0 of a line segment:

5 =L f /L 0 . 

The linear strain, e is

e =5-1.

In two dimensions, the stretch can be described by 

an ellipse as in figure B3 (Ramsay, 1967; Ragan, 1973; 

A.M. Johnson, unpub. data, 1981).

=(x 1 /S 1 ) (x 2 /S 2 )2 . (6)

In equation 6, the major and minor principal stretches 

are 5X and S2 , respectively. The clockwise angle between 

S1 and a given side of the triangle is a (fig. B3). The 

nondimensional coordinate jq is parallel to 51} and x2 is 

parallel to 52 . The radius of the circle deformed into the 

ellipse is equal to one.

The stretches within each triangular strain gauge 

(that is, each group of three stakes in the quadrilateral) 

are fully determined because values of stretch, 5, are 

known in three directions. In each triangle, Sly S2 , and a 

constitute three unknowns that can be determined by 

solving three equations relating the unknowns to the 

three stretches (see Appendix for solution of the 

equations).

Once Slt 52, and a are known, they may be used to 

compute the area strain, the finite shear strain -y, and the 

stretch, 5, in any direction (Jaeger, 1962; Ramsay, 1967; 

Ragan, 1973; A.M. Johnson, unpub. data, 1981). The 

area strain is d:

d =S 1 S 2 -1.

The maximum finite shear strain is ymax : 

Y«« =(51 2 -522 )/(25 1 5 2).

Tilt. Tilt is the difference between the initial and final 

amount of dip of any triangle in the quadrilateral. 

Similarly, change in dip direction is defined as the angle 

between the initial and final dip direction of a triangle.

lrThe quadratic elongation or quadratic extension, widely used 

in structural geology, is equal to the stretch squared (S2). For 

elongation, S is greater than 1, and for shortening, S is less than 1. Of 

course, the principal values of the right Cauchy-Green tensor are Sj2 

and S22.

Measurement of Slope Deformation B3
Ground 

unaffected

by 

landsliding

Landslide boundary

(strike-slip fault, showing 

relative horizontal movement)

\\

Landslide

B

Figure B2. Sketch of reference system for computing displacements across a discontinuity or landslide boundary. The x-y 

plane is horizontal; the origin is at point A and the positive x direction coincides with the direction of line AD. (A) Displacement 

component i/, is parallel to x, i/3 (not shown) is vertical, and uz is parallel to y. The angle between the horizontal projection of 

the total displacement and the x axis is X. (B) Initial position of point B is given by coordinates x, y, its final position, B', by 

coordinates x ', /'. The projection of angle DAB onto the x-y plane is <f>; that of angle DAB ' s '

Both are useful in interpreting strain data. Tilt shows 

whether the ground is steepening or flattening, and it is 

related to differences in the rate of thickening from one 

side of the quadrilateral to the other. Changes in the dip 

direction relate the tilt to changes in the shape of surface 

features.

The amount of dip and the dip direction of a 

triangle are computed as follows: The sides of a triangle 

correspond to coordinate directions £,- in the x-y plane, 

and the azimuths of those directions are w, (fig. B4). The 

slope (dz/d£i) and the azimuth (<y,) of each side of the 

triangle are computed from field data. The slope of a line 

in any arbitrary direction £, within a plane is computed by 

differentiating the equation of a plane,

z =ax +by,

with respect to £, and making the substitutions 

cos(«,-) =(dx/9&), and sin (w,-) =(9y/%):

(dz/d^ ) =a cos (wx ) +b sin (<a l ); (7a) 

B4 Landslide Processes in Utah

=acos(w2 ) (7b)

In these equations, a and b are constants describing 

the slope of the plane in the x and y directions, 

respectively. Using the slopes of two sides of the triangle, 

one solves the set of equations, 7a and 7b, simultaneously 

for a and b. The azimuth of the dip direction, which is the 

direction of maximum slope, is found by differentiating 

equation 7 with respect to w, letting the derivative of 

equation 7 equal zero, and solving for umax :

= tan (8)

The tangent of the dip angle, 6, is derived by substituting 

the azimuth of the dip direction, umax , back into equation 

7 and solving for (dz/di-max), the tangent.

and

tan (6) =a cos (umax ) +b sin (umax ), 

6 =tan-1 [a cos (umax ) +b sin (umax )].
NORTH

SAC

B

Figure B3. Diagram showing geometrical relationships 

between a deformed triangle and its stretch ellipse. (A) 

Triangle ABC in its final state. Angle Q'AB is angle CAB; angle 

9'ec is angle BCA. (B) The stretch ellipse. S1 and Sz are the 

principal stretches and are parallel to the xt and x2 

coordinates. SAB, SAC, and SBC are the components of 

stretch parallel to lines AB, AC, and BC, respectively. The 

angle is a ( =/? 'Ac ), the acute angle between line AC and 

xi> ft AB =9 AB ~ «; & BC = « +9'scRepeating these calculations for each triangle in its 

initial and final states and determining the differences 

gives four values of tilt and four values for the change in 

dip direction.

Figure B4. Diagram showing dip direction 

of triangle ABO, point A has the highest 

elevation. The positive x direction is due 

north. Line AB parallels coordinate direction 

£,, its bearing is «.,. Line AC parallels 

coordinate direction £2 , and its bearing is w2 . 

The dip direction parallels coordinate 

direction £ max, and its bearing is wmax.

Computation. The, data for computing the errors 

of closure, displacements, strains, and tilts are the slope 

distances and elevation differences between corners of 

the quadrilateral. The formulas for computing these 

quantities, given previously, were chosen to simplify the 

task of programming.

Separate programs are used for computing 

displacement or strain and tilt, and each program reads 

the data from files stored on a diskette. A file exists for 

each quadrilateral; each set of raw measurements in a file 

is identified by the date on which it was collected. 

Documentation and BASIC computer programs for 

creating the files and performing the computations are 

available from the U.S. Geological Survey (Johnson and 

Baum, 1987).

Field Technique

At each site selected for strain or displacement 

measurement, we drove four wooden stakes in the 

ground to mark the corners of a quadrilateral, and put a 

small finishing nail in the top of each stake (fig. Bl).

Measurement of Slope Deformation B5
We used a steel tape to measure distances between 

nails, hooking the end of the tape on one nail and reading 

the tape at the center of each of the other nails. The 

elevation of the top of each stake was obtained by taking 

backsights with a precision level. The azimuth of a 

reference line (diagonal AC) was measured with a Brunton compass so that strain and displacement data could 

be oriented.

No provisions were made to determine rigid body 

rotations of the quadrilaterals; where such data are of 

interest, accurate measurement of the azimuth of a 

reference line would be needed each time the length and 

elevation measurements were repeated. If the azimuth 

were known, additional computations would determine 

the rotation of a line that becomes parallel to the 

principal stretch.

We encountered three major difficulties in using 

the quadrilateral method: (1) Some of the stakes became 

tilted, loose, or broken, either by animals or heavy winter 

snow. (2) Repetition of length measurements showed 

that measurement errors of 3 mm were common and that 

errors of 5-6 mm sometimes occurred. (3) Errors of 0.5 

mm were fairly common in the elevation measurements, 

apparently because many stakes were tilted. However, 

errors were small in comparison to displacement.

In order to minimize the errors, we discounted 

results from stakes that were loose or greatly tilted. Also, 

we used the theory of quadrilaterals to detect errors of 

measurement of length.

We tried quadrilaterals of various shapes; however, 

most were approximately square (that is, equidimensional). Squares worked well because it was difficult to 

anticipate the direction of maximum strain at most 

locations on the landslide. The advantage of using a 

shape that is approximately square is that strain 

measurements are distributed throughout the four 

quadrants, whereas an elongate quadrilateral tends to 

concentrate strain measurements in two opposite 

quadrants.

Squares also worked well for displacement 

measurements. However, as quadrilaterals were 

distorted into diamond shapes, measurement errors 

increased. Thus, it was necessary to put new stakes in the 

ground from time to time to restore the original square 

shape of some of the quadrilaterals.

The quadrilaterals that we used ranged from 1.5 to 

10 m in width, but quadrilaterals that were from 1.5 to 4 

m wide seemed to work best for strain measurements. 

The larger quadrilaterals (from 6 to 10 m wide) were 

sensitive to inhomogeneous strain. Quadrilaterals 

smaller than 1.5 m would be too sensitive to 

measurement error.

Experience indicated that displacement 

measurements were needed more often, whereas strain 

measurements were needed less often than we made

them. During June and July 1983, and May 1984, the 

quadrilaterals were measured once every two or three 

days. Displacements averaged a few millimeters per day 

in 1983 and a few centimeters per day in 1984. By 

measuring every few days, a discontinuous record of 

displacements was obtained. However, the strain 

accumulating from a few days of movement was often too 

small to measure.

In order to satisfactorily measure displacement and 

strain in 1985, we measured the quadrilaterals once every 

three or four weeks and supplemented the quadrilaterals 

with three recording extension!eters along a flank of the 

landslide in order to obtain more nearly continuous 

displacement data. From the quadrilateral 

measurements, we computed the long-term (weeks or 

months) displacements at many points on the boundaries 

and long-term strains at points on the surface of the 

landslide. The combination of measurements used in 

1985 worked well for our purposes, except that 

erroneous measurements of strains were more difficult 

to detect.

Errors and Sensitivity

Detection and distribution of errors. When 

quadrilaterals were measured frequently (once every two 

to three days, or several times a day) many inaccurate 

data sets could be eliminated by inspection. 

Measurements that differed significantly from a series of 

otherwise constant values or a steady trend were 

regarded as inaccurate and were discarded. From the 

remaining sets of measurements, those having the 

smallest errors of closure were used for strain 

computations.

Errors of closure were computed by means of 

equation 1, as an aid in detecting inaccurate data sets. A 

simple method for distributing the errors, described by 

Laurila (1983, p. 41-45), was tried. In order to apply this 

method, we assumed that the length of one side of each 

quadrilateral, AB, was known accurately; however, we 

had no independent check on the length oiAB or any 

other length. Thus, if AB happened to be measured 

inaccurately, then using it to distribute the errors might 

make the adjusted measurements less accurate than the 

original measurements. Therefore, we decided to use the 

raw measurements, and to eliminate sets of 

measurements that had errors of closure larger than 0.30 .2 

By inspecting the raw measurements and checking

2By computing the error of closure for a square quadrilateral 

having sides exactly 2 m long, we found that an error of 6 mm in the 

measured length of one side caused an error of closure equal to 

approximately 0.3°.

B6 Landslide Processes in Utah
the magnitude of w, we eliminated data sets having 

closure errors larger than a few millimeters.

Sensitivity of measurements. When we measured 

quadrilaterals and then remeasured them a few minutes 

later, we found that our measurements could differ by as 

much as 5-6 mm, which is equivalent to a strain of 0.003 

in a quadrilateral with sides of 2 m length. Errors of 5-6 

mm occurred in 10-20 percent of our measurements. 

Consequently, the smallest strains that could be detected 

confidently by our measurements with a steel tape were 

0.003 (0.3 percent strain). As a result, the smallest strain 

that could be computed with an error less than 10 

percent was 0.030 (3.0 percent strain). We sometimes 

used measurements where the strains were less than 

0.030, especially if they were internally consistent and the 

orientations of the principal stretches for the four 

triangles were oriented in about the same direction.

The measurements were adequate for observing 

long-term strains. Strains of + 0.080 ( + 8.0 percent) to 

-0.180 (-18.0 percent) accumulated at many 

quadrilaterals during weeks and months of landslide 

movement. Thus, long-term strains were large compared 

to errors in the measurements.

Because boundary displacements of many 

decimeters accumulated as a result of days or weeks of 

landslide movement, measurement errors were small 

enough to disregard when computing displacements.

EXAMPLES OF QUADRILATERAL 

MEASUREMENTS

In order to illustrate how quadrilaterals have been 

used in the field to measure displacement, strain, and tilt, 

several examples from the Aspen Grove landslide in 

central Utah are cited: (1) the measurement of 

displacement at a landslide flank (strike-slip fault), (2) 

the measurement of displacement at a landslide toe 

(thrust fault), (3) the measurement of tilt and strain on a 

ridge, and (4) the measurement of strain at a place that 

unexpectedly became a normal fault. Before describing 

the history of the quadrilateral sites, we briefly describe 

the landslide.

The Aspen Grove Landslide

to 31°. From drilling and mapping in the vicinity of the 

landslide, we infer that the landslide debris occupies a 

trough in the surface of the bedrock.

The landslide debris upslope from a road that 

crosses the lower part of the landslide (fig. B6) is 

probably less than 10 m thick. At boreholes B-l, B-2, and 

B-3, in the upper part of the landslide, the debris is 6 m 

thick. Downslope from the lower road crossing, the 

debris may be as thick as 15-20 m beneath the large 

internal toe (between quadrilaterals Q-26 and Q-27, fig. 

B6).

The Aspen Grove landslide is 1 km long and its 

width changes gradually throughout its length. From the 

head (quadrilateral Q-l, fig. B6) to the vicinity of B-l, 

the width is from 50 to 55 m. Downslope from B-l, the 

width decreases to a minimum of 20 m near the 2340 m 

contour and then increases farther downslope to a 

maximum of 100 m on the internal toe. Downslope from 

the internal toe, the landslide is 70 m wide near 

quadrilateral Q-32, but beyond the edge of the mapped 

area, its width increases again to about 100 m.

Sliding at Aspen Grove began late in May 1983 

when a record snowpack containing from 75 to 100 cm of 

water melted in a few weeks. Discontinuous ground 

cracks marking the boundaries of the landslide were first 

discovered on May 31, 1983, and sliding continued for 

several weeks, until mid-July or early August. During this 

time, the landslide extended from the head at 

quadrilateral Q-l (fig. B6) to the vicinity of 

quadrilaterals Q-8 and Q-29. The landslide was dormant 

from late summer 1983 until approximately mid-April 

1984, when sliding was renewed in all parts of the 

landslide that had been active in 1983. Features at the 

surface of the landslide evolved dramatically. Many of the 

tension cracks in the head region opened wide, as the thin 

debris at the head slid downslope. A graben, near 

quadrilateral Q-l 3, and the flank ridge, at quadrilateral 

Q-2, enlarged. Flanks defined by en echelon cracks and 

fault segments in 1983 were defined by continuous 

strike-slip faults in 1984. The flanks of the landslide 

propagated farther downslope than in 1983. By the time 

we saw the landslide in mid-May, cracks had appeared at 

the 2,280 m contour on the internal toe. By the end of 

May, scattered cracks were visible downslope from the 

internal toe (near quadrilateral Q-32, fig. B6). In August 

1984, we observed a new shear zone that had started to 

form on the right flank3 , between quadrilateral Q-2 and 

quadrilateral Q-13.

The Aspen Grove landslide, on a northwest-facing, 

forested slope in Ephraim Canyon, central Utah (fig. B5), 

involves old landslide debris that consists of sandstone 

boulders in a sandy clay matrix. The slope angle of the 

landslide surface averages 11° and locally ranges from 0°

'The right flank is to the right of an observer standing at the 

crown and looking downslope. By this convention, right-lateral 

shearing occurs on the right flank.

Measurement of Slope Deformation B7
nr

ASPEN 

GROVE " 

LANDSLIDE

T.17S.

T.18S.

R.2E. R.3E. R.4E. 

Figure B5. Map showing location of the Aspen Grove landslide, central Utah.

Measurement of Displacement on a 

Landslide Flank

The flanks of the Aspen Grove landslide were 

characterized by strike-slip faults or shear zones. The 

faults were from a few centimeters to a few decimeters 

wide, so that small quadrilaterals worked well to measure 

displacement on the faults and strain parallel to the 

faults. The installation and history of quadrilateral Q-2 is 

described as an example of using a quadrilateral to 

measure displacement on a landslide flank.

Quadrilateral Q-2 was installed on the right flank, 

in the upper one-third of the Aspen Grove landslide, on 

June 15, 1983 (fig. B6). It was primarily intended to 

provide a sensitive measure of displacement on the right

flank. Also, the two stakes of Q-2 on moving ground 

could be used to measure one-dimensional strain 

adjacent to the strike-slip fault on the right flank.

A perspective sketch (fig. B7), based on a 

photograph, shows the site as it appeared on the day the 

stakes were installed. Two cracks marked the location of 

a right-lateral shear zone. The shear zone was parallel to 

the axis of a low ridge, from 50 to 80 cm high, and passed 

through the northeast flank of the ridge.

Small swales with axes parallel to the shear zone 

occurred on each side of the ridge. The one on the outer 

side of the ridge, on stationary ground, is shown in the 

foreground in figure B7. It was about 2 m wide and 1 m 

deep; it had a smooth, roughly semicircular profile. The 

swale on moving ground was subtle; it was marked by a 

small furrow at the boundary between the ridge and the

B8 Landslide Processes in Utah
100 METERS

EXPLANATION

\

Strike-slip fault Showing relative horizontal movement

Toe or thrust fault Sawteeth on up-

" per plate

y"' Scarp Hachures on downdropped side

- '/ Cracks in surface of landslide

 jg-^Topographlc contour

B-1 -f Borehole 

Q-6 Quadrilateral

area 'b'" " "X .. /'

Figure B6. Map showing the Aspen Grove landslide, August 1984.

slightly arched, convex surface of the landslide. Twenty 

meters farther downslope, both swales became more 

prominent and the ridge became a conspicuous 

topographic feature.

Stakes C and D were placed on stationary ground 

northeast of the ridge so that line DC was parallel to the 

shear zone, which had a trend of N. 41° W. Stakes^ and 

B were placed on the crest of the ridge, on moving 

ground, a few decimeters southeast of the shear zone 

(opposite stakes C and D).

The size, orientation, and shape of the 

quadrilateral were determined mainly by the geometry of 

the site. The quadrilateral needed to be large enough to 

straddle the shear zone and allow clearance for the 

measuring tape to stretch between the tops of the stakes. 

Had stakes A and B been located off the crest of the 

ridge, clearance might have been a problem. The 

reference line (line CD, the line defined by stakes on 

stationary ground) was approximately parallel to the 

shear zone so that the largest component of displacement

would be parallel to the reference line. This orientation 

was chosen for convenience in making the measurements 

and in orienting the displacement vectors. The 

trapezoidal shape was a result of trying to fit the 

quadrilateral to the site.

During part of June and July 1983, the 

quadrilateral was measured once every two or three days, 

and displacement on the shear zone was computed for 

the period from June 15 to August 4. Movement on the 

shear zone amounted to 14 cm between June 15 and July 

10, when movement ended. The azimuth of the 

displacement was N. 49° W., or 8° counterclockwise from 

the trend of the shear zone. Thus, the stakes were 

displaced toward the center of the landslide, away from 

the shear zone. The quadrilateral functioned very well. 

The strike-slip displacements were much larger than the 

measurement errors.

Renewed movement began in the spring of 1984, 

and displacement of 2.2 m, nearly parallel to the trend of 

the fault zone, occurred between August 4, 1983, and

Measurement of Slope Deformation B9
Tree

Figure B7. Sketch of the site of Q-2 (part of area "a", shown 

in fig. B6) made from photograph taken by R.W. Fleming on 

June 15, 1983. Ground surface depicted by means of serial 

profiles in perspective view.

May 23, 1984. As a result of the displacement, the shear 

zone defined by en echelon cracks in 1983 was replaced 

by a through-going fault in 1984 (fig. B8). Moist, lightbrown, slickensided clay was exposed at the ground 

surface on either side of the fault, and vegetation was 

disrupted in a strip, from 10 to 20 cm wide, along the 

fault.

Shortening of 1.2 percent occurred between stakes 

A and B on the low ridge within the landslide between 

August 1983 and May 23, 1984. Shortening was 

surprising because the maximum rate of displacement 

was occurring approximately 230 m downslope from 

Q-2, and we had measured progressively less 

displacement in several places between this point of 

maximum displacement and Q-2. Thus, we expected the 

strain at Q-2 to be extensional.

As a result of the movement during the spring of 

1984, stake B had moved into a position where the tape 

was deflected by the ground surface when measuring 

between stake B and stake C or D. Therefore, the 

movement of stake B was overestimated in our 

calculations. No action was taken to correct this situation 

during 1984, because we were able to measure to point A 

without difficulty.

The landslide was active through June and July and 

stopped moving sometime during August 1984. We 

measured Q-2 once every two or three days between 

May 23 and June 7, and once on August 3. Between May 

23 and August 3, 1.6 m of displacement occurred on the 

fault.

Displacement was oriented 11° counterclockwise 

from the trace of the fault and plunged only 1.5°. The 

shallow plunge was interesting because one might expect 

the movement to be roughly parallel to the ground 

surface, which sloped 10°. The nonzero component of 

displacement of stakes normal to the fault, toward the 

center of the landslide, was also interesting because we 

observed no evidence of dilation near the fault. Both of 

these, though, might reflect a secondary flow of the 

landslide debris, which we associate with the construction 

of lateral ridges.

By April 1985, quadrilateral Q-2 had been 

deformed into a diamond-shaped figure. Therefore, after 

measuring the distances between all of the stakes, stakes 

A and B were removed and replaced with new stakes in 

positions that were across the shear zone and opposite 

stakes C and D.

During the period that quadrilateral Q-2 was in 

use, the site changed. The shear zone evolved from a set 

of en echelon cracks to a through-going strike-slip fault.

The quadrilateral measured displacement of 

several meters, with sensitivity of several millimeters. 

Even though the quadrilateral was small, 2 m by 2 m, it 

was able to measure displacements amounting to a few 

meters before stakes A and B had to be replaced in order 

to restore the original shape of the quadrilateral.

Measurement of Displacement at a 

Landslide Toe

In May 1984, we observed that the Aspen Grove 

landslide was enlarging distally. In 1983, it had 

terminated a few tens of meters below the lower road 

crossing (fig. B6). In 1984, cracks were appearing farther 

downslope, toward a large internal toe of the landslide 

that had formed during a previous episode of movement 

(near quadrilateral Q-27, fig. B6). We expected the 

internal toe to be reactivated and become the toe of the 

modern Aspen Grove landslide; we wanted to observe 

the early stages of movement as the toe became 

reactivated. Quadrilateral Q-27 was installed about 

halfway across the width of the toe on May 18, 1984, in 

order to accomplish this.

The internal toe was one of the most conspicuous 

features of the landslide surface (fig. B9A). The slope 

angle of the profile of the distal end of the toe was about 

24°, and the angle gradually reduced to 16° about 40 m 

upslope (fig. B9B). The toe was about 7 m high, and its 

maximum width was about 100 m. However, the 

landslide was narrower upslope from the toe (fig. B6).

A site for a quadrilateral midway across the toe was 

chosen in an area relatively clear of vegetation that would 

interfere with measurement. Movement was expected to 

be greatest in this area; therefore, it was the optimum

B10 Landslide Processes in Utah
10 METERS

NORTH

CONTOUR INTERVAL 1 METER

EXPLANATION

Strike-slip fault Showing relative horizontal 

movement

Topographic contour

Thrust fault Sawteeth 

on upper plate

Stake at corner of quad* 

rilateral

/ / y Narrow cracks In surface 

' ' ' of landslide

Q-2 Quadrilateral

 wv Wide cracks in surface 

of landslide

Figure B8. Map showing part of the Aspen Grove landslide in vicinity of quadrilateral Q-2 during August 1984 (part of area 

"a", shown in fig. B6).

Measurement of Slope Deformation B11
CONTOUR INTERVAL 1 METER 

EXPLANATION

Thrust fault Sawteeth on upper plate

Cracks in surface of 

landslide

Topographic contours

B m

Stake at corner of 

quadrilateral

Direction of landslide 

movement

Q-26 Quadrilateral

METERS 

2300 -

2275

 Ground surface

Limiting locations of failure surfaces) associated with toe

Landslide debris

Figure B9. Map and cross section showing internal toe of the Aspen Grove landslide (area "b", shown in fig. 6). (A) Map view, 

August 1984. Landslide movement is from left to right. (B) Cross section Z-Z'. Dotted lines show locations of two possible slip 

surfaces.

place to detect small movements. Also, the toe was well 

defined topographically near its middle, so it seemed 

possible to estimate within a meter or two where the 

failure surface would intersect the ground surface.

The quadrilateral was oblong, from 2.2 to 2.8 m

wide and 4.4 m long, and its long dimension was parallel 

to the axis of the landslide. Line CD was installed on 

gently sloping ground, downslope from the toe (fig. B9). 

Points C and D were a meter downslope from where the 

thrust was expected to intersect the ground surface. We

B12 Landslide Processes in Utah
tried to make line CD parallel to the local trend of the 

toe, so that motion would be approximately 

perpendicular to line CD. Points A and B were 4.3 m 

upslope from line CD, and were 1 m higher in elevation 

than stakes C and D.

Within the quadrilateral, the ground surface sloped 

gently (a few degrees) for 1 m upslope from stakes C and 

D. The slope then increased abruptly to 20°. The slope 

was relatively smooth, but pebbles, cobbles, and small 

undulations made the slope locally rough.

In order to document differential movement 

occurring in the toe, we measured Q-27 every few days 

from May 18 to June 6, 1984. The computed 

displacements for stake A are shown in figure BIO. 

Although we measured displacements of 6 cm between 

May 18 and June 2, no physical changes could be 

observed in the site.

When we returned to the site on June 6, we 

observed a band of cracked soil on the face of the toe, just 

northeast of line AD. The displacement between May 18 

and June 6 was 9 cm. We do not know how much 

movement occurred in the toe before measurements 

were started at Q-27, but we do know that displacement 

of at least 6 to 9 cm was needed before the ground cracks 

became visible.

In August 1984, the cracks observed in June were 

no longer visible, but other thrust features appeared 

along the toe. Three branches of the thrust were 

observed and mapped within the boundaries of the 

quadrilateral (fig. Bll). These changes were associated 

with 22 cm of displacement between May 18 and August 

4, 1984.

Points A and B, which were on the toe upslope 

from the thrust features, moved approximately parallel to 

the axis of the landslide. During May 1984, displacement 

of point A was nearlyliorizontal. The ground surface of 

the landslide below Q-27 slopes about 7°, so the 

displacement at Q-27 has an upward component relative 

to the ground surface. Later displacements, occurring 

between June 6 and August 4, had a vertical upward 

component (fig. BIO).

These movements were relative to stakes C and D, 

which themselves were moving on a landslide element 

that was downslope from the internal toe. Thus, the 

magnitude and direction of movement of stakes A and B 

relative to nonmoving ground is unknown.

Measurement of Strain and 

Tilt on a Ridge

In June 1984, it was observed that the ridge near 

Q-2, on the right flank of the landslide, was deforming. 

Q-2 was at the upslope end of the ridge. Most evidence 

for growth of the ridge was 15 to 20 m farther downslope. 

Thus, stakes were installed there to measure and 

document the deformation that was occurring. Several 

quadrilaterals were installed on the ridge, one of which 

(Q-34) is described here.

In 1983, very few cracks were present within the 

boundaries of the Aspen Grove landslide. On June 2, 

1984, near the beginning of the second season of 

movement, cracks were observed in the crest of the small 

ridge on the right flank of the landslide, near Q-2. Large 

tension cracks, oblique to the axis of the ridge, were

b

|

u

z

i

60

40-

20-

18B

Displacements of stake A Q-27

Cracks forming

F~f

2 6

__ - ~" *~~ " Total displacement 

"""" __ __ «._ ...I * Vertical displacement

MAY JUNE JULY AUGUST

Figure B10. Graph showing displacements of stake A at Q-27 from May 18 to August 4, 1984.

Measurement of Slope Deformation B13
2282

2279

2278

NORTH CONTOUR INTERVAL 1 METER

5 METERS 

J

EXPLANATION

Thrust fault Sawteeth 

on upper plate

Topographic contour

Stake at corner of 

quadrilateral

0-27 Quadrilateral

Figure B11. Enlarged view of area "d", shown in figure B9. 

Three thrust faults were noted within the boundaries of 

quadrilateral Q-27 in August 1984.

observed at several places along the ridge crest. A fallen 

tree trunk lying on the south flank of the ridge had been 

lifted out of a depression it had formed in the ground 

surface.

The ridge itself was approximately 1-1.2 m high, 3 

m wide, and 20 m long (fig. B6; fig. B12). In cross section, 

the ridge was rounded and slightly asymmetric. The north 

side of the ridge, which was parallel to the strike-slip 

fault, was steeper and higher than the south side of the 

ridge. The surface of the ridge was covered with boulders 

and had a thin layer of top soil.

Quadrilateral Q-34 was one of several points of 

measurement intended to document the growth and 

deformation of the ridge. One longitudinal and two 

transverse lines of stakes were surveyed on the ridge and 

across to stable ground. Also, two quadrilaterals were 

placed on different parts of the ridge. Quadrilateral Q-34 

was intended to monitor a simple, relatively smooth 

section of the inside flank of the ridge. Two of the stakes, 

C and D, were also part of a transverse stake line across 

the ridge. The transverse line provided data on ridge 

growth, and Q-34 provided data for computation of 

strain and tilt. Stakes A and B also provided a check on 

the results from stakes C and D. We did not assume that 

growth of a ridge is two-dimensional.

Stakes A and D were installed near the crest of the 

ridge so that line AD was nearly parallel to the axis of the 

ridge. Stakes B and C were installed downslope from 

stakes A and D, near the base of the south side of the 

ridge, so that the four stakes defined the corners of a 

trapezoid (fig. B12). The quadrilateral was 3.3 m long and 

2.6 m wide, and side AD was 0.5 m higher than side BC.

Measurements were made on June 2, June 5, June 

7, and August 3,1984. Strain and tilt from June 2 to June 

7 were small, and displacement along the right flank at 

Q-2 was only 0.15 m. However, the magnitudes and 

directions of the strains were consistent among the four 

triangles in the quadrilateral. The maximum principal 

strain had a small positive value and was oriented toward 

the west or northwest. The minor principal strain 

indicated small amounts of shortening or elongation. 

Minor amounts of tilt, oriented toward the south, also 

occurred.

Considerably more deformation had occurred at 

Q-34 by August 3. The ridge was displaced 0.68 m 

between June 7 and August 3 (measured at Q-2), 

making a total displacement of 0.83 m since 

measurements started on June 2. The major principal 

strain at Q-34 for the period June 2 to August 3 was 

oriented toward the west-northwest and was between 

+1.3 and +2.5 percent (table Bl). The minor principal 

strain was from -0.3 to -0.6 percent. Continued tilting 

toward the south-southwest resulted in the southwest 

side of the ridge being 1.0-1.5° steeper than it had been

B14 Landslide Processes In Utah
* \

NORTH

5 METERS

CONTOUR INTERVAL 1 METER

Strike-slip fault Showing 

relative horizontal movement

Thrust fault Sawteeth on 

upper plate

Narrow cracks in surface 

-^ of landslide Arrow shows 

y \ direction of movement of 

landslide

EXPLANATION

Wide cracks in surface of 

landslide

Anticline Showing direction 

of plunge

Overturned, doubly plunging 

anticline

Monocline

Topographic contour

Stake at corner of 

quadrilateral

Q-34 Quadrilateral

Figure B12. Map showing ridge on the right flank of the Aspen Grove landslide (part of area "a", shown in fig. B6) (August 

1984). Quadrilateral Q-34 was located on the southwest side of the ridge, on upper part of the landslide.

on June 2 (fig. B13). The site of the quadrilateral looked During 1985, deformation of the ridge continued,

about the same as it had before, except that the cracks The principal strains for the period June 2, 1984 to

just upslope from Q-34 were slightly wider than they had August 2, 1985, indicate that elongation in a west or

been on June 2. west-northwest direction continued. A small amount of

Measurement of Slope Deformation B15
Table B1. Principal strains at Q-34

Beginning Final Triangle Major Azimuth - Major Minor 

Date Date Principal Principal Principal 

Strain Strain Strain 

(percent) (percent)

6-2-84 8-3-8A .ABC +1.58 N. 81° W. -0.76 

BCD +1.30 N. 63° w. -0.32 

CDA +2.31 N. 62° W. -0.33 

DAB +2.51 N. 73° W. -0.55

6-2-84 8-2-85 ABC +5.91 N. 73° W. -1.91 

BCD +6.82 N. 89° w. +0.48 

CDA +4.62 N. 86° W. +0.82 

DAB +3.74 N. 66° W. -0.91

DIP, IN DEGREES 

20 15 10 5 0

^5o 50 N, / f^^i \ 

CS} 40\. / / /

a. ABC ''+4 3^^-^J 

b. BCD <% 20 ;~ 

c. CDA **# 10 

d. DAB 

SOL

5 

10 

15

20

TH

Although shortening of the quadrilateral perpendicular 

to the ridge axis would have produced similar results, the 

leveling data show that the vertical distance between line 

AD and line BC actually increased. For example, the 

height differences between stakes A and B increased 

from 0.48 m on June 2, 1984 to 0.54 m on August 4, 1984.

The ridge elongated approximately parallel to its 

axis. Deformation perpendicular to the axis of the ridge 

was small, and shortening occurred in some triangles but 

elongation occurred in others. Area strain was positive 

(increasing area) throughout the period of activity; 

however, no tension cracks formed within the 

quadrilateral. Positive area strain is normally associated 

with thinning of the landslide debris. However, positive 

area strain in Q-34 was associated with growth in the 

height of the ridge.

Figure B13. Polar plot showing attitude (dip and dip 

directions) of the four triangles in quadrilateral Q-34. Solid 

dots, attitudes on June 2, 1984; open circles, attitudes on 

August 3,1984; and solid triangles, attitudes on August 2, 

1985.

shortening occurred in triangles ABC and DAB and 

elongation occurred in triangles BCD and CDA in a 

north or north-northeast direction (table Bl). Tilt 

increased at three of the triangles and decreased slightly 

at one (fig. B13). The 1985 deformation resulted from 1.36 

m of displacement on the old flank fault (2.19 m for 1984 

and 1985 combined) and 1.95 m of displacement on the 

new flank fault.

The measurements of the quadrilateral showed 

that the ridge was, in fact, growing in height while the 

landslide was moving during 1984 and 1985. Steepening 

of the side of the ridge, as documented by the increasing 

tilt of the quadrilateral, showed that the ridge was 

growing in height relative to the neighboring ground.

Area of Developing Normal Faults 

Within the Landslide

During early June 1983, only a few cracks were the 

sole evidence of deformation anywhere within the 

boundaries of the Aspen Grove landslide. Quadrilateral 

Q-5 was set on a topographic high point near the axis 

(midway between the two flanks) of the landslide, at a 

high point along a prominent change in slope. East of 

Q-5, the surface of the landslide had a backslope of a few 

degrees toward the northeast (fig. B6). West of Q-5, the 

surface of the landslide sloped toward the northwest, 

roughly parallel to the direction of sliding, but the ground 

was locally steepened.

We expected purely extensional strains at Q-5 

because the most rapidly moving part of the landslide was 

170 m downslope from Q-5. Although we wanted to 

observe these strains, we were perhaps more interested 

in monitoring changes in what was initially a high point in 

the surface of the landslide.

B16 Landslide Processes in Utah
Quadrilateral Q-5 was 1.8 m wide, 2 m long, and 

roughly rectangular. The long dimension of the 

quadrilateral was oriented N. 50° E. and the short 

dimension was oriented N. 30° W. Thus, the sides of the 

quadrilateral were slightly oblique to the axis of the 

landslide, which was oriented N. 55° W. The 

quadrilateral sloped gently (about 8°) toward the northnorthwest.

During 1983, quadrilateral Q-5 remained 

unchanged. The computed strains were of the same size 

as those that would result from the expected errors in the 

measurements (table B2) and the principal strains showed 

inconsistent orientations among the four triangles. Thus, 

any deformation that occurred during 1983 was too small 

to be detected by our measurements.

During 1984, it was apparent that Q-5 no longer 

stood on the high point of the landslide. The highest point 

in the area was now upslope from Q-5 and closer to the 

left flank (fig. B14). Tilt of quadrilateral Q-5 increased 

from August 4, 1983 to May 23, 1984. From May 23 to 

August 3,1984, the slope of line AB increased while the 

slope of line AC remained unchanged and the slope of 

line AD decreased. Thus, the quadrilateral sloped from 

10° to 11° toward the north-northwest by August 3,1984.

Strains were relatively small in 1984 and the 

principal strains were difficult to interpret; however, the 

deformation can be understood by looking at the relative 

motions of the four stakes (table B2). For the period from 

August 4,1983 to May 23,1984, the strains in individual 

stake lines show that stakes A and C moved toward stake 

B. Between May 23 and August 3,1984, stakes.4, C, and 

D maintained their positions relative to one another, but 

stake B moved away from the other three.

In 1985, the difference in topographic form 

became even more pronounced than it had been in 1984 

(fig. B15). Tension cracks that had been noted near the left 

flank in 1984 appeared upslope from and within 

quadrilateral Q-5. By this time, Q-5 had changed from a 

gently sloping quadrilateral on a high point to a steeply 

sloping quadrilateral downslope from a high point. The 

quadrilateral sloped from 11° to 12° toward the 

northwest or north-northwest. Line AB continued to 

steepen during 1985, while lines AC and AD became 

flatter.

During the period from August 3,1984 to August 

4, 1985, the major principal strain was oriented between 

west and northwest, or roughly parallel to line BD. The 

minor principal strain indicated a small amount of 

contraction normal to that direction. As in the latter part 

of 1984, the relative motions of individual stakes 

indicated that triangle ACD remained relatively undeformed, while staked moved away from it (table B2). Such 

motion is consistent with the appearance of cracks 

between stake B and triangle A CD during 1985.

In 1986, cracks that appeared during 1985 

developed into a normal fault (fig. B16) that dropped stake 

B 13 cm relative to staked between August 2,1985 and 

September 12, 1986. The fault scarp appeared to be 

between 10 and 15 cm high and nearly vertical. 

Horizontal displacement of stake B relative to line AD 

amounted to only one centimeter during the same 

period; thus, movement was roughly parallel to the fault 

surface exposed in the scarp. Stakes C and D moved 

upward relative to stake A so that triangle ACD was 

relatively flat in September 1986.

Deformation at quadrilateral Q-5 was surprising. 

While the first few meters of movement occurred, the 

quadrilateral deformed very little but steepened as it 

apparently moved off a high point in the slip surface of 

the landslide. Crack-like deformation began during 1984 

as stake B moved away from triangle A CD, well before 

cracks appeared within the quadrilateral during 1985. 

What appeared to be simple tilting of the quadrilateral 

during 1984 and 1985, apparently was a precursor of the 

movement on the normal fault that occurred in 1986. 

Thus, had we recognized the style of deformation evident 

in the data, we could have predicted the formation of a 

normal fault passing through Q-5 long before the fault 

could be observed in the field.

DISCUSSION AND CONCLUSION

We have described a method for measuring the 

displacements, strains, and tilts at the surface of a 

landslide and have illustrated the field technique by 

means of examples.

The method is practical and can be easily done 

along with field mapping and other data-collection 

activities. The field technique is simple and the required

Table B2. Strains in Quadrilateral Q-5

Beginning

Date

6-16-83

8-4-83

5-23-84

8-3-84

Final

Date

8-4-83

5-23-84

8-3-84

8-2-85

Strain in lines (percent)

AB

0.0

-1.4

+0.3

+4.5

BC

+0.1

-0.7

+0.3

+1.1

CD

+0.1

+0.9

-0.2

-0.1

DA

-0.2

+0.7

+0.2

-0.1

AC

-0.1

-0.3

-0.3

-0.5

BD

+0.1

0.0

+0.6

+3.5

Measurement of Slope Deformation B17
NORTH

10 METERS

CONTOUR INTERVAL 1 METEfl

EXPLANATION

Strike-clip fault Showing 

S/t relative horizontal movement

Thrust fault Sawteeth on 

upper plate

Narrow cracks in surface 

of landslide

Wide cracks in surface 

of landslide

st*"* Scarp Hachures on

downdropped side C 

Topographic contour Q 5

Borehole

Stake at corner of 

quadrilateral

Quadrilateral

Figure B14. Map showing part of the Aspen Grove landslide in the vicinity of Q-5 during August 1984 (area "c", shown in 

fig. B6).

measurements can be made rapidly. In the heavily The equations provided in the text have general

wooded terrain of the Aspen Grove landslide, two people application because they treat three-dimensional 

could measure 28 quadrilaterals and eight stake lines in displacements and large strains. Thus, the equations are

one working day. more useful than graphical methods (Ragan, 1973; TerB18 Landslide Processes in Utah
10 METERS

CONTOUR INTERVAL 1 METER

NORTH

EXPLANATION

Strike-slip fault Showing 

relative horizontal movement

Thrust fault Sawteeth on 

upper plate

Narrow cracks In surface 

of landslide

Wide cracks in surface 

of landslide

Scarp Machines on 

downdropped side

Topographic contour

81

B

Q-5

Borehole

Stake at corner of 

quadrilateral

Quadrilateral

Figure B15. Map showing part of the Aspen Grove landslide in the vicinity of Q-5 during August 1985 (area "c", shown in 

fig. B6).

Stepanian and Ter-Stepanian, 1974). BASIC computer 

programs are available for processing the data (Johnson 

and Baum, 1987).

When we installed the first few quadrilaterals, we 

hoped to measure strains and displacements in a simple 

way. As data accumulated, it was obvious that strains and

displacements could be measured, and we began asking 

questions that required more data and better precision in 

the measurements. A few simple improvements would 

have resulted in a better data set.

We should have used something sturdier than 13 by 

40 mm (nominal 1 by 2 ) wooden stakes and should

Measurement of Slope Deformation B19
NORTH

CONTOUR INTERVAL 1 METER

EXPLANATION

10 METERS 

J

Strike-slip fault Showing 

relative horizontal movement

Thrust fault Sawteeth on 

upper plate

Narrow cracks in surface 

of landslide

Wide cracks in surface B1 

of landslide

Scarp Hachures on /

downdropped side B

Topographic contour Q-5

Borehole

Stake at corner of 

quadrilateral

Quadrilateral

Figure B16. Map showing part of the Aspen Grove landslide in the vicinity of Q-5 during September 1986 (area "c*, shown 

in fig. B6).

have used metal tags with numbers stamped into them to 

identify the stakes. Several alternatives are available, 

from sturdier wooden stakes to elaborate concrete and 

metal stakes having machined heads that accept tape or 

rod extensiometers.

During years when the landslide was visited 

infrequently, we should have replicated measurements 

during each visit in order to eliminate large errors that 

resulted from careless mistakes. Measurements were 

made with a tape that was marked in feet and inches, thus

B20 Landslide Processes in Utah
it was easy to make mistakes when reading the tape or 

recording the measurement. Errors in measurements 

caused by such mistakes were frequently large enough to 

interfere with strain computations; thus, repeated 

measurements would have improved the quality of our 

data sets.

Replication of measurements and application of 

tension and temperature corrections to measurements 

made with a steel tape would have improved sensitivity to 

small strains. Greater accuracy and sensitivity could have 

been achieved by making measurements with a tape 

extensiometer or rod extensiometer (Franklin, 1984; 

Hanna, 1985) so that strains as small as 0.0001 (0.01 

percent strain) could » be computed accurately. 

Quadrilateral measurements obtained by such means 

would be amenable to correction by least squares (Laurila, 1983) or Ingrain's (1972) method of angular value 

weighting.

We should have replaced the stakes on moving 

ground next to strike-slip faults at the end of each season 

to keep the quadrilaterals as nearly equidimensional as 

possible, because measurement errors increased when 

the quadrilaterals became stretched into narrow 

diamond-shaped figures.

We should have taken photographs, and made 

sketches and detailed maps to show how some of the sites 

changed through time. The magnitudes of strains and 

displacements that result in visible changes to the ground 

surface are poorly known. Carefully drawn sketches and 

large-scale maps of the quadrilateral sites might have 

been even more useful than the photographs, because 

they would have insured that we studied the sites 

carefully and thoughtfully.

Planning is an important aspect of the field 

technique. We tried to view each quadrilateral as an 

experiment that had at least one specific question to be 

answered. Observations made during a visit to the site 

usually prompted questions. Quadrilaterals were then 

designed and surveyed in order to answer these 

questions. In some cases, we may not have asked 

significant questions, but we always learned something 

from the quadrilaterals as we measured and observed 

changes in them.

Once we had questions in mind, we tried to 

position and configure the quadrilaterals so that they 

would give optimum response to the deformations. Thus, 

quadrilaterals used to measure displacements were 

positioned at several locations on the boundaries and 

they were configured so that two sides were 

approximately parallel to the discontinuity at the 

boundary. Likewise, quadrilaterals used to measure 

strain and tilt were positioned on deformational features 

that were poorly understood or in uncomplicated areas 

where the background pattern of deformation could be

observed. Quadrilaterals used to measure strain and tilt 

were configured and sized to fit the feature being 

measured.

During the course of measuring and observing the 

quadrilaterals, our understanding of the landslide 

increased and new questions presented themselves. As 

these questions arose, we evaluated the adequacy of our 

quadrilaterals to answer the new questions. In some 

cases, the new questions could be answered with more 

quadrilaterals; in other cases, the questions could be 

answered only by other types of measurements or 

observations.

APPENDIX

Because analytic expressions for computing the 

magnitudes and directions of principal stretches from the 

stretches in three arbitrary directions are not available in 

any standard references, we present a brief derivation.

Equation 6, in the text, can be rewritten using the 

notation of figure B3:

(1/SAB2 ) ^(l/^2 ) cos2 (ft,,)

+ (l/522) sin2 (^B ), (Ala)

(l/SBC2 ) = (l/Si 2 ) cos2 03flc )

+ (l/S22 )sin2 08BC ), (Alb)

and

(1/^c2 ) = (I/Si 2 ) cos2 

+ (l/S2 2 )sin2 (&4C ). (Ale)

Equations Ala, Alb, and Ale can be rewritten by 

using trigonometric identities that relate cos (2/3) to 

cos2 (j8 ) and sin2 (/3 ), and by making the substitution (fig. 

B3):

(1/SAB2 ) = 1 2 +l/522]

S - 1/S22 ] cos 2 (a + Q'AB )}, (A2a) 

Measurement of Slope Deformation B21
and

(1/SBC 2 ) =(1/2){[1/51 2 + 1/S22 ]

21 ~, I cos 2 (a J v

(l/SAC2 ) = + 1/S22 ]

[1/S1 2 -1/S22 ] cos (2a). (A2c)

/AOU\ (A2b) v ;

Because a is the angle between the final orientation of 

AC and the major principal stretch direction, the 

angles 9 AB and 9 BC are computed for the final shape

of the triangle.

AXV c- j- u * -.,. * * >> After finding a, we substitute a into equations 3a

and 3b and rearrange them to get 5j and S2 :

- 1/^c2 ) (1 - cos (2a))

(c°s - cos 2(a ; (A6)

Subtracting equations 2b from 2a, and A2c from 2a, 

yields:

1/SAB2 - 1/SBC 2 - US 2 ) {cos 2(o +Q'AB )

-cos2(a (A3a)

1/SAB 2 - IISAC2 = (US 2 - US 2 )

{cos 2(a + Q'AB )- cos (2a )}. (A3b) 

Dividing equation 3a by 3b yields:

USAB2 - USBC2 = cos 2 (a + Q'AB ) -cos 2 (a +0 BC) 

cos 2 (a + Q'AB ) - cos (2a). - 1/SAC2

(A4)

We use the angle addition formula for cosines to isolate 

terms containing a and rearrange equation 4 to solve 

for a. Thus,

a = (1/2) tan'1 {[(cos (2Q'AB ) -1)

(1/SAB2

/[sin

(1/SAB2 - l/SBC2 y 

AC2 ) +cos (29 Vc ) - cos (2Q'AB )] 

(1/SAB2 - 1/SBC2)/ (1/SAB 2 - l/SAC2)

+ sin (20 'BC) - sin (20 '

S = +cos

/(cos 2(« +0'XB ) - cos 2(« +0 Bc))}(1/2) . (A7)

In the event that all four corners of a quadrilateral 

are coplanar, a least-squares method can be used to find 

a single set of best-fit values for Slt S2 , and a that apply 

to the entire quadrilateral. Equations 2a, 2b, and 

2c are simplified by applying the angle-addition 

formula:

2/S.2 = (US 2 + US22 ) + (US 2 - US22 ) 

cos (2a) cos (20 ',-)

- (US 2 - US22) sin (2a) sin (20',-). (A8)

We have an equation of the form:

z c +CiX +c 2 y 

in which

c =(US 2 +l/522 ), 

Ci =(l/5!2 - l/522 ) cos (2a), 

>i 2 - l/52 2 ) sin (2a).

(A5)

and

Applying stnnd.ird methods for least squares results in 

three equations lhat can be solved for c, c^ and c2 and, 

hence, a, 51} and S2 .

B22 Landslide Processes in Utah
REFERENCES CITED

Brinker, R.C., and Wolf, P.R., 1977, Elementary surveying, in 

the collection The IEP series in civil engineering, 6th 

edition: New York, Thomas Y. Crowell Co., 568 p.

Brunsden, D., 1984, Mudslides, in Brunsden, D. and Prior, 

D.B., eds., Slope instability: New York, John Wiley, p. 

363^18.

Carter, M., and Bentley, S.P., 1985, The geometry of slip 

surfaces beneath landslides Predictions from surface 

measurements: Canadian Geotechnical Journal, v. 22, p. 

234-238.

Cruden, D.M., 1986, The geometry of slip surfaces beneath 

landslides Predictions from surface measurements

 Discussion: Canadian Geotechnical Journal, v. 23,

p. 94. 

Franklin, J.A., 1984, Slope instrumentation and monitoring, in

Brunsden, D., and Prior, D.B., eds., Slope instability:

New York, John Wiley, p. 143-170. 

Hanna, T.H., 1985, Field instrumentation in geotechnical

engineering, in Series on Rock and Soil Mechanics, v. 10:

Rockport, Maine, Trans Tech Publications, 843 p. 

Ingram, F.L., 1972, The Ingram method of angular value

weighting in trilateration: Surveying and Mapping, v. 32,

p. 509-515. 

Iverson, R.M., 1984, Unsteady, nonuniform landslide motion-

 Theory and measurement: Stanford, Calif., Stanford 

University Ph.D. thesis, 303 p.

Jaeger, J.C., 1962, Elasticity, fracture, and flow, 2nd edition: 

London, Methuen, and New York, John Wiley, 208 p.

Johnson, A.M., and Baum, R.L., 1987, BASIC programs for 

computing displacements, strains, and tilts from 

quadrilateral measurements: U.S. Geological Survey 

Open-File Report 87-343, 19 p.

Keefer, O.K., and Johnson, A.M., 1983, Earth flows Morphology, mobilization, and movement: U.S. Geological 

Survey Professional Paper 1264, 56 p.

Lantz, J.R., 1984, Geology and kinematics of a clay-rich 

landslide with an undulatory slip surface: College Station, 

Texas, Texas A & M University M.S. thesis, 132 p.

Laurila, S.H., 1983, Electronic surveying in practice: New York, 

John Wiley, 388 p.

Malvern, L.E., 1969, Introduction to the mechanics of a 

continuous medium: Englewood Cliffs, New Jersey, 

Prentice Hall, 713 p.

Ragan, D.M., 1973, Structural geology, an introduction to 

geometrical techniques: New York, John Wiley, 208 p.

Ramsay, J.G., 1967, Folding and fracturing of rocks: New York, 

McGraw -Hill, 568 p.

Ter-Stepanian, G., 1984, Analysis of slope deformation for 

determining the landslide mechanism, in Proceedings of 

the Fourth International Symposium on Landslides, v. 2: 

Downview, Ontario, Canada, University of Toronto 

Press, p. 499-504.

Ter-Stepanian, G., and Ter-Stepanian, H., 1974, Determination 

of deformed state of a sliding body by creep hodographs, 

in Proceedings of the Second International Congress of 

the International Association of Engineering Geology: 

Sao Paulo, Brazil, v. II, p. V23.1-V23.5.

Zaruba, Q., and Mencl, V., 1982, Landslides and their control, 

2nd revised edition: New York, Elsevier, 324 p.

Measurement of Slope Deformation B23

*U.S. GOVERNMENT PRI NT1 NG OFFICE: 1988-0-673-047/86023



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