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52-18 The Civil Engineering Handbook, Second Edition

Hence, from Eq. (52.46), 0.007953 £ P(G

) £ 0.011375. Performing similar computations for other

joint probabilities and applying Eq. (52.45) yields the second-order bounds as 0.0367 £ p

£ 0.0447.

52.5 Reliability-Based Design

There are two principal considerations in the design of structures: the ﬁrst is the optimization of the

total expected utility of the structure by the designer, and the second is the optimization of the design

code by the controlling authority. In the latter, the optimization covers as much as possible the range of

practical structures and includes issues such safety, serviceability, and overall cost. A complete practical

design code will have to be simple, yet consider all the above factors for the whole family of structures

and account for many details, some of which may not have been fully tested. As such, it is natural that

code formats and requirements evolved over a long period of time and collective wisdom and consensus

of the profession remain signiﬁcant factors. Such a process is seen to be less formal and has obvious

drawbacks when new materials, structural principles, and technological developments are introduced.

Nevertheless, previous generation codes still play a major role as guides to new generation codes in the

process known as code calibration.

Reliability concepts presented in the earlier sections serve as a plausible vehicle for code calibration,

where various limit state conditions such as ultimate and serviceability limit states need to be addressed.

Based on the calibration results, it is possible in principle when formulating the new code to incorporate

the type of failure and the associated consequences of failure through a cost component. To enforce such

requirements explicitly will make the codiﬁed design procedure for general use unnecessarily too complex

a process in the present-day context. Nevertheless, it can be incorporated in the code in a less obvious

manner through the numerical factors speciﬁed in the code. Reliability analysis is being accepted currently

as a practical, consistent, and formal tool for which partial factors can be derived given the safety-checking

formats in code formulation.

Load and Resistance Factor Design Format

Before outlining the code calibration and formulation procedure, it is appropriate to brieﬂy mention

safety-checking formats. The format for codes of practice differs between countries, and there are attempts

to unify such format. For example, all European structural design codes follow the general form speciﬁed

by the Comite Europeen du Beton (CEB, 1976). Numerous partial factors for both materials and loads

are imposed to account for various uncertain components. The Canadian building codes adopted a less

complicated format by using only one partial factor to account for material uncertainties (NRCC, 1977).

The AISC code in the U.S. uses the load and resistance factor design (LRFD) format, expressed as

(Ravindra and Galambos, 1978)

(52.51)

where f and g

are the resistance and load factors, respectively, R

the nominal resistance, and S

the

mean load effects.

Code Calibration Procedure

Changes or improvements to existing codes can result for reasons of harmonization of different codes

or development of a simpler code. The ﬁrst comprehensive change toward reliability-based design code

()

FF1 8167

2 2123 0 8718 1 8167

108718

0 003422

0 007953

...

fgRS

nkkm

Structural Reliability 52-19

formulation in the United States was the American National Standard A58 on minimum design loads in

buildings (Ellingwood et al., 1980). Calibration has been deﬁned as the process of assigning values to the

parameters in a design code to achieve a desired level of reliability accounting for practical constraints,

resulting in a speciﬁc design code. It involves the combination of judgment, ﬁtting to existing design

practice, and optimization.

The ﬁrst step in the code calibration procedure is to deﬁne the scope of applicability, namely, the class

of structures (e.g., bridges), materials (e.g., concrete), failure modes, geographical domain of validity,

and geometrical properties. This will result in selecting the range of values for the design variables, such

as length, cross-sectional areas, permitted yield stress, and applied loads. For practical computation,

discrete zones of such values are used and the corresponding frequency of occurrence corresponding to

each zone in actual practice estimated. For example, the dead-to-live load ratios in building structures

are usually conﬁned within the range of 0.5 to 2.

In the second step, for each zone the existing structural design code is used to design the various

elements, such as beams and columns, with speciﬁed geometry and load conditions.

The third step involves the speciﬁcation of performance functions. The limit state conditions to be

speciﬁed in the new code must be deﬁned, such as those related to ﬂexure, shear, local buckling, and

deﬂection. The performance function for each limit state is expressed in terms of the basic variables.

In order to compute the reliability index corresponding to each performance function, in the fourth

step, the statistics of the basic variables are assumed to be available. These are often compiled from extensive

survey data. An example of statistical data on resistance can be found in Galambos et al. (1982) and is

reproduced in Table 52.5.

The ﬁfth step involves computing the reliability indices of each member designed under the old code

using techniques such as FORM. An example of the reliability indices obtained from such a process is

shown in Fig. 52.11, reproduced directly from Galambos et al. (1982). It shows the range of reliability

indices implied in existing code for gravity loads. If the range is unacceptably wide, further work is needed

to obtain a new set of factors satisfying the intended goals.

The speciﬁcation of the goal of the revised or new code is a major step of the process of calibration.

The overall goal could be to maximize the expected utility or to achieve a speciﬁed failure probability or

reliability index taking into consideration the failure consequences. The target reliability index to be used

for a revised code can be based on the values obtained in the previous step. For example, a statistically

determined value can be chosen, such as the weighted average or some speciﬁed percentile value of b

TABLE 52.5 Summary of Statistical Data on Resistance

Designation R/R

Probability

Distribution

Reinforced concrete, ﬂexure

Grade 60

Grade 40

1.05

1.14

0.11

0.14

Normal

Reinforced concrete, short-tied columns 0.95 0.14 Normal

Reinforced concrete beams, shear, minimum stirrups 1.00 0.19 Normal

Structural steel

Tension members, yield

Compact beam, uniform moment (plastic design)

Beam-column (plastic design)

1.05

1.07

0.11

0.13

0.15

Lognormal

Cold-formed steel, braced beams 1.17 0.17 Lognormal

Aluminum, laterally braced beams 1.10 0.08 Lognormal

Unreinforced masonry walls in compression,

inspected workmanship 5.3 0.18 Lognormal

Glulam beams

Live load

Snow load

1.97

1.62

0.18

Weibull

Source: From Table 1 of Galambos, T.V. et al., J. Struct. Div. ASCE, 108, 959, 1982.

52-20 The Civil Engineering Handbook, Second Edition

based on the occurrence frequency of the various designs encountered in practice. For example, Siu et al.

(1975) obtained an average b value of 4.67 for hot-rolled steel columns under compression failure for

the 1975 NBCC code. It is unlikely that a single value will be selected as the target for an entire code.

For example, failure consequence is a major design consideration, and higher values of b are often

imposed for brittle-type failure. To achieve the desired goal, a quantitative measure or objective function

must be formulated for example (Ditlevsen and Madsen, 1996),

(52.52)

where w

are the relative frequencies of occurrences (determined in the ﬁrst step). The function M(.) can

be simply in terms of reliability indices, such as (b

* – b

)

where b

* is the target reliability indices. This

function can take on a more complex form by introducing cost variables (including cost of initial

construction and cost of failure consequences).

For a given code safety-checking format, the set of partial factors is determined based on the objective

function. First, trial values of partial factors are used for the new format and the respective b

computed.

The objective function is computed using Eq. (52.52). By repeating for different sets of values for the

partial factors, the set of values that minimize Eq. (52.52) are adopted. The same procedure is repeated

for the various load combinations to be implemented in the new (or revised) code.

Evaluation of Load and Resistance Factors

As an illustration of the computation of the load and resistance factors, a simple example involving only

the dead and live loads is considered. The new code format is speciﬁed as

(52.53)

where R

, D

, and L

are the nominal resistance, dead load, and live load, respectively, and f, g

, and

are the corresponding partial factors. Consider the case of a steel member under tension with the

performance function expressed as

(52.54)

The statistics of the basic variables are given in Table 52.6. For the speciﬁc class of loads corresponding

to a mean live-to-dead load ratio

–

—

D of 1.5, the partial factors are to be computed for a target reliability

index of 3.0.

FIGURE 52.11 Reliability index for steel and reinforced concrete beams. (From Galambos, T.V. et al., J. Struct. Div.

ASCE, 108, 959, 1982.)

0 0.5 1.0 1.5 2.0 2.5 3.0

Typical range for

reinforced concrete

Typical range

for steel

or S

= 400 ft

Curve Description R/R

RC - Grade 60 D + L

RC - Grade 40 D + L

RC - Grade 60 D + S

RC - Grade 40 D + S

Steel D + L

Steel D + S

1.05

1.15

1.05

1.15

1.07

0.11

0.14

0.11

0.14

0.13

()

ÂÂÂ

failure

parameters

modes

materials

fg gRDL

nDnLn

≥ +

gRDLX

()

=--

Structural Reliability 52-21

The computation can be summarized as follows:

1. To commence the computation, the most likely failure point is assumed, say r* = 0.866R

, d* =

1.03D

and l* = 1.99L.

2. The equivalent normal parameters for R and L are next computed. For R, which is lognormal

distributed,

For L, which follows the type I extreme distribution,

3. Based on the performance function and the target reliability index,

4. The sensitivity factors at the most likely failure point is given by the direction cosines and computed

TA BLE 52.6 Statistics of Basic Variables

Va riables R/R

Coefﬁcient of

Va riation, V

Probability

Distribution

Resistance, R 1.05 0.11 Lognormal

Dead load, D 1.05 0.10 Normal

Live load, L 1.00 0.25 Type I extreme

s V

rr V

()

-- +

()

** .

ln ln

0 09497

0 9854

uL D

== =

=-- =

5 1302 3 4201

0 5772

1 3312

Fl l u

fl l u lu

Lnn

Lnnn nn

exp exp

exp

()

=-- -

()

{}

[]

()

=--

()

-- -

()

{}

[]

aa a

0 9965

0 01191

lFlDs

()

[]

()

[]

0 8821

0 606

mmm

sss

--==

RRDD RD

222

0 8898 3 9491 4 5137 0 4 6482

.. . . or

sss

222

0 4452

52-22 The Civil Engineering Handbook, Second Edition

5. The most likely failure point is then updated,

6. If the results in step 5 are signiﬁcantly different from the values assumed in step 1, then the values

in step 5 are used for the next iteration. Steps 2 to 5 are then repeated until convergence. The ﬁnal

solution of the most likely failure point is used to estimate the partial factors.

7. Assuming the values given in step 5 are correct, the mean partial factors of safety are given by

The nominal partial factors of safety are

Hence the code checking equation can be written as

The implied overall mean factor of safety can be estimated as

whereas the overall nominal factor of safety is approximately 1.80.

Acknowledgment

The author expresses his gratitude to Dr. Kok-Kwang Phoon for reading through the manuscript and

providing constructive suggestions.

References

Ang, A.H.-S. and Tang, W.H., Probability Concepts in Engineering Planning and Design, Vol. I, Basic

Principles, John Wiley & Sons, New York, 1975.

Ang, A.H.-S. and Tang, W.H., Probability Concepts in Engineering Planning and Design, Vol. II, Decision,

Risk, and Reliability, John Wiley & Sons, New York, 1984.

sss

222

0 1009

0 8897

rRRR

lDL

*.....

*. .

=- = - * *=

masb0 9854 0 4452 0 09497 3 0 0 8586

1 030

2 960 1 9736

0 859

1 030

1 974

=*=

...

0 859 1 05 0 902

1 030 1 05 1 082

1 974 1 0 1 974

090 108 197...RDL

nnn

≥ +

DL+

≥

()

103 1974 0 859

186

.. .

Structural Reliability 52-23

Bjerager, P., Reliability Analysis of Structural Systems, Series R, No. 183, Department of Structural Engi-

neering, Technical University of Denmark, Lyngby, 1984.

CEB, Common Uniﬁed Rules for Different Types of Construction and Material, Bulletin d’Information

116-E, Comite Europeen du Beton, Paris, 1976.

Der Kiureghian, A. and Liu, P.L., Structural reliability under incomplete information, J.Eng. Mech. ASCE,

112, 579, 1986.

Ditlevsen, O. and Madsen, H.O., Structural Reliability Methods, John Wiley & Sons, New York, 1996.

Ellingwood, B. et al., Development of a Probability Based Load Criterion for American National Standard

A58, NBS Special Publication SP577, National Bureau of Standards, Washington, D.C., 1980.

Galambos, T.V. et al., Probability based load criteria: assessment of current design practice, J. Struct. Div.

ASCE, 108, 959, 1982.

Greig, G.L., An assessment of high-order bounds for structural reliability, Struct. Safety, 11, 213, 1992.

Harbitz, A., Efﬁcient and accurate probability of failure calculation by the use of importance sampling

technique, in Applications of Statistics and Probability in Soil and Structural Engineering, Augusti, G.,

Borri, A., and Vannuchi, O., Eds., Pitagora Editrice, Bologna, 1983, p. 825.

Ma, H.F. and Ang, A.H.-S., Reliability Analysis of Redundant Ductile Structural Systems, Civil Engineer-

ing Studies, SRS 494, University of Illinois at Urbana-Champaign, 1981.

Madsen, H.O., Krenk, S., and Lind, N.C., Methods of Structural Safety, Prentice Hall, Englewood Cliffs,

NJ, 1986.

Melchers, R.E., Simulation in time-invariant and time-variant reliability problems, in Reliability and

Optimization of Structural Systems, Rackwitz, R. and Thoft-Christensen, P., Eds., Springer, Berlin,

1991, p. 39.

Melchers, R.E., Structural Reliability: Analysis and Prediction, John Wiley & Sons, Chichester, U.K., 1999.

Nowak, A.S. and Collins, K.R., Reliability of Structures, McGraw-Hill, New York, 2000.

NRCC, National Building Code of Canada, National Research Council of Canada, Ottawa, 1977.

Quek, S.T. and Ang, A.H.-S., Structural System Reliability by the Method of Stable Conﬁguration, Civil

Engineering Series, SRS 529, University of Illinois at Urbana-Champaign, 1986.

Rackwitz, R. and Fiessler, B., Structural reliability under combined random load sequences, Comput.

Struct., 9, 489, 1978.

Ravindra, M.K. and Galambos, T.V., Load and resistance factor design for steel, J. Struct. Div. ASCE, 104,

1337, 1978.

Siu, W.W., Parimi, S.R., and Lind, N.C., Practical approach to code calibration, J. Struct. Div. ASCE, 101,

489, 1975.

Related Journals and Conferences

Although papers on structural reliability can be found in numerous journals and conference proceedings,

the author wishes to draw attention to two civil engineering-related journals and two conferences. These

are the Journal of Structural Safety, the Journal of Probabilistic Engineering Mechanics, the International

Conference on Structural Safety and Reliability, and the International Conference on the Applications

of Statistics and Probability. In addition, there are numerous specialty conferences organized in this ﬁeld.

VII

Surveying

Engineering

Edward M. Mikhail

Purdue University

53 General Mathematical and Physical Concepts

Edward M. Mikhail

Coordinate Systems

•

Plane Geometry

•

Three-Dimensional Geometry

•

Ve c t or Algebra

•

Matrix Algebra

•

Coordinate Transformations

•

Linearization of Nonlinear Functions

•

Map

Projections

•

Observational Data Adjustment

54 Plane Surveying

Steven D. Johnson and Wesley G. Crawford

Introduction

•

Distance Measurement

•

Elevation Measurement

•

Angle Measurement

•

Plane

Survey Computations

•

Horizontal Curves

•

Ve rtical Curves

•

Vo l ume

Geodesy

B. H. W. van Gelder

Introduction

•

Coordinate Representations

•

Coordinate Frames Used in Geodesy and Some

Additional Relationships

•

Mapping

•

Basic Concepts in Mechanics

•

Satellite Surveying

•

Gravity Field and Related Issues

•

Reference Systems and Datum Transformations

56 Photogrammetry and Remote Sensing

J. S. Bethel

Basic Concepts in Photogrammetry

•

Sensors and Platforms

•

Mathematics of

Photogrammetry

•

Instruments and Equipment

•

Photogrammetric Products

•

Digital

Photogrammetry

•

Photogrammetric Project Planning

•

Close-Range Metrology

•

Remote

Sensing

57 Geographic Information Systems

Jolyon D. Thurgood and J. S. Bethel

Introduction

•

Geographic Information Components

•

Modeling Geographic Information

•

Building and Maintaining a GIS

•

Spatial Analysis

•

Information Extraction

•

Applications

•

Summary

URVEYING IS ONE OF THE OLDEST ACTIVITIES of the civil engineer, and remains a primary

component of civil engineering. It is also one ﬁeld that continues to undergo phenomenal changes

due to technological developments in digital imaging and satellite positioning. These modern

surveying tools are not only revolutionizing regular surveying engineering tasks but are also impacting

a myriad of applications in a variety of ﬁelds where near-real-time positioning is of great value.

Surveying and engineering are closely related professional activities. The area of surveying and mapping

is in many countries a discipline by itself, and taken in total, it is almost as broad in scope as civil

engineering. In the U.S., surveying engineering has been historically allied to civil engineering. Engineer-

ing surveying is deﬁned as those activities involved in the planning and execution of surveys for the

VII

-2

The Civil Engineering Handbook, Second Edition

location, design, construction, operation, and maintenance of civil and other engineered projects. Such

activities include the preparation of survey and related mapping speciﬁcations; execution of photogram-

metric and ﬁeld surveys for the collection of required data, including topographic and hydrographic

data; calculation, reduction, and plotting of survey data for use in engineering design; design and

provision of horizontal and vertical control survey networks; provision of line and grade and other layout

work for construction and mining activities; execution and certiﬁcation of quality control measurements

during construction; monitoring of ground and structural stability, including alignment observations,

settlement levels, and related reports and certiﬁcations; measurement of material and other quantities

for inventory, economic assessment, and cost accounting purposes; execution of as-built surveys and

preparation of related maps and plans and proﬁles upon completion of construction; and analysis of

errors and tolerances associated with the measurement, ﬁeld layout, and mapping or other plots of survey

measurements required in support of engineering projects. Engineering surveying may be regarded as a

specialty within the broader professional practice of engineering and, with the exception of boundary,

right-of-way, or other cadastral surveying, includes all surveying and mapping activities required to

support the sound conception, planning, design, construction, maintenance, and operation of engineered

projects. Engineering surveying does not include surveys for the retracement of existing land ownership

boundaries or the creation of new boundaries.

Modern surveying engineering encompasses several specialty areas, each of which requires substantial

knowledge and training in order to attain proper expertise. The most primary area perhaps is plane

surveying because it is so widely applied in engineering and surveying practice. In plane surveying, we

consider the fundamentals of measuring distance, angle, direction, and elevation. These measured quan-

tities are then used to determine position, slope, area, and volume — the basic parameters of civil

engineering design and construction. Plane surveying is applied in civil engineering projects of limited

areal extent, where the effects of the earth’s curvature are negligible relative to the positional accuracy

required for the project.

Geodesy, or higher surveying, is an extensive discipline dealing with mathematical and physical aspects

of modeling the size and shape of the earth, and its gravity ﬁeld. Since the launch of earth-orbiting

satellites, geodesy has become a truly three-dimensional science. Terrestrial and space geodetic measure-

ment techniques, and particularly the relatively new technique of satellite surveying using the Global

Positioning System (GPS), are applied in geodetic surveying. GPS surveying has not only revolutionized

the art of navigation but has also brought about an efﬁcient positioning technique for a variety of users,

prominent among them the engineering community. GPS has had a profound impact on the fundamental

problems of determining relative and absolute positions on the earth, including improvements in speed,

timeliness, and accuracy. It is safe to say that any geometry-based data collection scheme proﬁts to some

degree from the full constellation of 24 GPS satellites. In addition to the obvious applications in geodesy,

surveying, and photogrammetry, the use of GPS is applied in civil engineering areas such as transportation

(truck and emergency vehicle monitoring, intelligent vehicle and highway systems, etc.) and structures

(monitoring of deformation of structures such as water dams). Even in other areas such as forestry and

agriculture (crop yield management) GPS provides the geometric backbone of modern (geographic)

information systems.

Photogrammetry and remote sensing encompass all activities involved in deriving qualitative and

quantitative information about objects and environments from their images. Such imagery may be

acquired at close range, from aircraft, or from satellites. In addition to large-, medium-, and small-scale

mapping, many other applications such as resource management and environmental assessment and

monitoring rely on imageries of various types. Close-range applications include such tasks as accident

reconstruction, mapping of complex piping systems, and shape determination for parabolic antennas.

Large-scale mapping (including the capture of data on infrastructure) remains the primary civil engi-

neering application of photogrammetry. Recent evolution toward working with digital imagery has

brought about the increasing acceptance of the digital orthophoto to augment or supplant the planimetric

map. Digital image processing tools offer the probability of great increases in mapping productivity

through automation. For small- and medium-scale mapping, the increasing availability of satellite image

Surveying Engineering

VII

-3

data offers an alternative to chemical photography. Commercially available satellite data with spatial

resolutions of 1 to 3 meters, proposed for the near future, would have a profound impact on all mapping

activities within civil engineering. Inclusion of GPS in photogrammetric and remote sensing acquisition

platforms will lead to substantial improvements in accuracy, timeliness, and economy.

For centuries, maps have provided layered information in graphical form and have been used as legal

documents and as tools to support decision making for applications such as urban planning. Recently,

geographic information systems (GIS) have broadened the role played by all types of maps to encompass

a total system of hardware, software, and procedures designed to capture, manage, manipulate, and

produce information in a spatial context. GIS applications are broad indeed; they include land record

management, base mapping, infrastructure maintenance, facilities management, and many others.

A driving force behind the move toward integrating mapping and other spatially oriented data has

been the various utility industries and municipalities who need to plan and manage their infrastructure

facilities and property assets. This automated mapping/facilities management, or AM/FM, concept is

being used successfully today by many cities, counties, and utility industries, who may have embarked

on the transition as much as 15 years ago. Successful practitioners of GIS can satisfy the needs of a broad

spectrum of users with a single system, minimizing the duplication of resources required to support

historically independent user groups.

All other components of surveying engineering contribute to the construction of a GIS. The range of

survey methods, from classical to modern geodesic and space-based technologies, provide the required

reference framework. Digital mapping provides an efﬁcient technology to populate the GIS with spatial

information. Remote sensing techniques applied to the earth and its environment provide the various

thematic layers of information.

Scope of This Section of the Handbook

The scope of Section VII, Surveying Engineering, in this handbook is to present the reader with the basic

information involved in the performance of different surveying engineering projects. As was mentioned

earlier, this is a discipline of many areas, each of which will be covered in a separate chapter. The

underlying mathematical concepts used by the different areas of surveying engineering are covered ﬁrst

in Chapter 53, followed by four chapters covering, in sequence, plane surveying, geodesy, photogram-

metry and remote sensing, and geographic information systems. Of particular importance is the topic

on measurements, their errors, and least squares adjustment of redundant data. Since surveying is

fundamentally a measurement science, all phases are covered: preanalysis (design), data acquisition

(observations), data preprocessing, data adjustment, and postadjustment analysis of the results (quality

assessment). Each engineering surveying project must properly execute these phases.

General Mathematical

and Physical Concepts

53.1 Coordinate Systems

Two-Dimensional Systems

•

Three-Dimensional Coordinate

Systems

53.2 Plane Geometry

Distance between Two Points

•

Slope of a Line between Two

Points

•

Azimuth

•

Bearing

•

Relation between Azimuth and

Slope Angle

•

Internal and External Division of a Line Segment

•

Equation of a Line with Known

and

Intercepts

•

General

Equation of a Line

•

Lines Parallel to the Axes

•

Equation of a

Line with Given Slope and

Intercept

•

Equation of a Line with

a Given Slope Passing through a Given Point

•

Equation of a

Line Joining Two Points

•

Equation of a Line with Given Length

and Slope of the Perpendicular from Origin

•

Perpendicular

Distance from the Origin to a Line

•

Perpendicular Distance

from a Point to a Line

•

Equation of a Line through a Point

and Parallel to Another Line

•

Equation of a Line through a

Point and Perpendicular to Another Line

•

Angle between Two

Lines

•

Point of Intersection of Two Lines

•

Equation of a Circle

•

Intersection of a Line and a Circle

•

Areas

53.3 Three-Dimensional Geometry

Distance between Two Points

•

Equation of a Plane

•

Equation

of a Straight Line

•

Equation of a Sphere

53.4 Vector Algebra

Deﬁnitions

•

Ve c tor O p erations

•

Planes and Lines

53.5 Matrix Algebra

Deﬁnition

•

Types of Matrices • Basic Matrix Operations •

Matrix Inverse

53.6 Coordinate Transformations

Linear Transformations

•

Nonlinear Transformations

53.7 Linearization of Nonlinear Functions

One Function of Two Variables

•

Tw o Functions of One Variable

•

Two Functions of Two Variables Each

•

General Case of

Functions of

Va r iables

•

Differentiation of a Determinant

•

Differentiation of a Quotient

53.8 Map Projections

53.9 Observational Data Adjustment

Mathematical Model for Adjustment

•

Design/Preanalysis

•

Data Acquisition

•

Data Preprocessing

•

Data Adjustment

•

Least

Squares Adjustment

•

Te c hniques of Least Squares

•

Assessment

of Adjustment Results

•

Conﬁdence Region for Estimated

Parameters • Applications in Surveying Engineering

Edward M. Mikhail

Purdue University