03 MeasurementScience W

7/25/2019 03 MeasurementScience W

1/39


2/39

ERE 371

Measurement Science Introduction

Principles apply to any activity relying on measurements

Considerations

Surveying measurements !servations


3/39

ERE 371

vervie"

#asic principles of measurements

Su!se$uent use of measurements

Planning for measurements %nits of measurements


4/39

ERE 371

Measurements

Single measurement

Issues


5/39

ERE 371

Measurements

Met&ods for increasing confidence

E'pressions of $uality


6/39

ERE 371

Precision

%nderstanding precision

(egree of precision depends on

E'press precision using statistics


7/39

ERE 371

)ariance and Standard (eviation

Purpose

Indicate measurement spread

*ormulae

( )1n

''s++

= +ss=

E'ample

Set ,- +.3 m/ +.0 m/ +. m2 s+ 4.41 m+/ s 4.1 m Set #- +.1 m/ +.3 m/ +. m 2 s+ 4.40 m+/ s 4.+ m

standard deviation-variance-


8/39

ERE 371

,ccuracy

%nderstanding accuracy

E'ample ,ccepted value is +.31 m

5&ic& measurement set is more accurate/ , or #6

Set , average- +.0 m

Set # average- +.3 m


9/39

ERE 371

,CC%R,C

PRECISI8

9ig& :o"

9ig&

:o"


10/39

ERE 371

E'pressing ,ccuracy

,!solute accuracy

Relative accuracy


11/39

ERE 371

Considering Measurement )ariations

Issue

;"o approac&es

Sources of error ;ypes of error


12/39

ERE 371

;&eory of Errors

;&eory

Error

Issues


13/39

ERE 371

#asic Principles

*our underlying principles in ;&eory of Errors

Confidence is provided !y comparing to


14/39

ERE 371

;&eory of !servations

;&eory

Issues


15/39

ERE 371

Errors and Mista


16/39

ERE 371

;ypes of Errors

Systematic errors =!iases>

Random errors


17/39

ERE 371

Redundancy in Measurements

?enerally ma

Residuals


18/39

ERE 371

%nderstanding Random Errors

*re$uency of occurrence

(ispersion

*re$uency

9istogram

8ormal distri!ution


19/39

ERE 371

:a"s of Pro!a!ility

:a"s of pro!a!ility

Increased precision (ecreased precision


20/39

ERE 371

E'pressing Precision

E'pressing precision using standard deviation

Interpreting standard deviation

Percent errors

%se percent errors to specify re$uired precision

E@4 1.A00@ E@ 1.@@@ E.g. re$uire E@to !e less t&an specific value

( )

1n

''s

+

=

1n

rs

+

=


21/39

ERE 371

Readings

C&apter 3 sections 3.1 3.1A


22/39

ERE 371

,pplying Measurement Science

Su!se$uent %se of Measured )alues

Preanalysis of Measurements

Importance of %nits


23/39

ERE 371

Su!se$uent %se of Measured )alues

;ypes of o!servations

ften use measurements to compute values

;&ree specific ideas


24/39

ERE 371

Significant *igures

;"o types of numeric values

In calculations

Issue


25/39

ERE 371

%se of 5eig&ts

,pplications

Incorporate multiplicative factors into computation

)ery po"erful tool


26/39

ERE 371

5eig&ted Measurements

Situation- &ave greater confidence in some values

9o" to determine "eig&t6

Commonly use "eig&t derived !y inverse proportion

+factor

1por

factor

1p ==


27/39

ERE 371

5eig&ted Measurements E'ample

3.373

=0.+7>=3.44>=+.B3>MMeanRegular =

++==

?roup1

?roup+

?roup3

+.0 +.B 0.

3.4 3.4 3.+

3.1 3.+ .1

Sum B. @.4 1+.B

Mean +.B 3.4 0.3

S.(. 4.3B 4.+4 4.@7


28/39

ERE 371

5eig&ted Measurements E'ample

3.44=1.4A>=+.44>=A.@A>

7>=1.4A>=0.+44>=+.44>=3.=+.B3>=A.@A>M" =

++

++=

?roup1

?roup+

?roup3

+.0 +.B 0.

3.4 3.4 3.+

3.1 3.+ .1

Sum B. @.4 1+.B

Mean +.B 3.4 0.3

S.(. 4.3B 4.+4 4.@7

5eig&t A.@A +.4 1.4A

+i =s.d.>

1p =


29/39

ERE 371

Propagation of )ariance

)ariance of o!servations

Issue

)ariance of result depends on


30/39

ERE 371

Propagation of )ariance =Contd>

,ssumptions

;&eoretical formulation

( )D/E//F/f% =

( ) ( ) ( ) ( )+D+

+

+

+

+

+

F

+

+

% GD

%G

%G

%G

F

%G

++

+

+

=

8ote- Standard deviation %found !y ta


31/39

ERE 371

Propagation of )ariance E'ample

Situation

Measure slope distance and vertical angle

Calculate &oriHontal distance

Measurements made Slope distance =:> 1+0.4A7 m "it& S.(. =s:> 4.417 m

,ngle => 31 4J +K "it& S.(. =s> 44 44J 14K 4.44440B radians

Calculating &oriHontal distance 9 : cos 1+0.4A7 m ' cos =31 4J +K> 14A.+0 m


32/39

ERE 371

Propagation of )ariance E'ample

9o" good is calculated value of 96 %se propagation of variance to estimate variance

Calculating variance Lcos:9 =

mMm4.BA3>4J+K=31cosLcos:

9===

mradNA0.4AA71>4J+K=31sin1+0.4A7NL>=Nsin:L

9===

+++++

L

+

+

:

+

9 >14=0.B=NA0.4A7>=4.417>=4.BA>sL

9s:

9s

+=

+

=

4.41m4.410B@s9 ==


33/39

ERE 371

Propagation of )ariance =Contd>

*ormulas in te't

?eneraliHing propagation of variance


34/39

ERE 371


Issue

#asically inverse of propagation of variance

Steps


35/39

ERE 371


Situation Reducing slope distance to &oriHontal

9o" "ell does vertical angle need to !e measured6

Steps

(ifferentiate 9 ".r.t.

ScosL9 =

(ivide !y original e$uation

Ignore sign/ rearrange to get d=radians>

SsinLdLd9 =

dLtanLScosL

dLSsinL

9

d9=

=

( ) ( )9tanL

d9dL=

dLSsinLd9 =


36/39

ERE 371


E'ample

Measure slope distance along O1 incline

5ant relative accuracy in 9 of 1 in 14/444

9o" "ell do "e need to measure 6

( ) ( )radians4.444373

14/444tan1

1dL =

=

( ) ( )9tanL

d9dL=

41J17K477K7A.@KdL

= Implication

8eed a 1J instrument


37/39

ERE 371

%nits of Measurement

:inear distance units

9istoric- ?unters c&ain- 1 c&ain AA feet 0 rods 144 lin


38/39

ERE 371

%nits of Measurement

,rea units

Englis&- ,cre- 1 acre 1 c&ain ' 14 c&ains 03A4 ft+

Metric- 9ectare- 1 &a 144m ' 144m 14/444 m+

)olume units Englis&- Cu!ic yards/ ,creNfeet

Metric- Cu!ic meters

;&e %.S. and t&e metric system


39/39

Documents

03 MeasurementScience W