Comp Ens

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Compensarea re retelelor libere de trilateratie prin masuratori

indirecte

Plecam de la relatia v(n,1) = A(n,n)x(u,1)+l(n,1) P(n,n)

Nx+ATPl=0 (1)

PlANx T1--=

Compensarea retelelor libere

De regula, la compensarea retelelor geodezice se dau anumiteelemente fixe:

v Coordonatev Orientariv Baze, etc

In functie de aceste elemente retelele geodezice se impart in:

Retele libere acele retele la care lipsesc in totalitate sau paertialaceste elemente;

Retele legate la care se dau numarul strict de elementenecesare;

Retele constranse acele retele in care avem elementesupraabundente;

Rtelele libere sunt acele retele in care lipsesc in totalitate sau partialelementele fixe, necesarepentru a incadra reteaua intr-un anumit sistem .
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Numarul de elemente fixe reprezinta defectul de date initiale (d-defectul

de datum), si este functie de natura retelei; d poate lua urmatoarele

valori: (tabel)

Dimensiunea

retelei

Denumire

(natura retelei)

Nr. maxim

d

Elemente

necesare

1 nivelment 1

Translatia pe Z

Tz

2 planimetrie 4

Translatii Tx, Ty

Orientare

Factor de scara:m

3

G.P.S.

Retele

combinate 3D

7

Translatii Tx,Ty, Tz

Rotatii x, y,z

Scara:m

Cteodat apare i defectul de configura ie, dac suntem pu i nimposibilitatea dterminarii coordonatelor punctelor retelei.

Fie reteaua de trilateratie din figura de mai jos:

8

1

42 6

3 5 7
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In patrulaterul 1,2,3,4 este bine conformat; in patrulaterul ,in patrulaterul

urmator lipseste o diagonala, iar in ultimul lipseste o masuratoare.

De acel defect de configuratie trebuie sa se tina seama la aplicarea

testelor statistice post procesare.

Defectul total: dt=d+dk d-defect de datum

dk-defect de configuratie

Intr-o retea libera s-au executat n observatii (determinari) , se cere sa

determine valorile cele mai probabile ale parametrilor si preciziile cu

care au fost determinati acesti parametri.

Dependenta functionala intre observatii si parametri este data derelatia :

M+v=F(x) modelul fuunctional (1)

modelul stochastic masuratorile sunt independente

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Determinarea coeficienilor ecuaiilor empirice

prin metoda celor mai mici ptrate

Consider m un set de date experimentale (xi yi), i= n,1 pentru care dorim sadetermin m o func ie care s descrie ct mai bine dependen a dintre

variabilele ii yix MM

Func ia se alege fie dup alura graficului sau dup anumite considera ii

teoretice. De obicei, aceast dependen se consider ca o func ie

polinomial .

....2 +++= cxbxay

Dac se consider un polinom de gradul I vom avea ca necunoscute

bxay += ( 1 )

iii bxavy +=+ ( 2 ),

unde vi este corec ia ce modific ecuaia dreptei

Y

trasarea

V1

V2V3

t1 t2 t3

Y=f(x)

t(x)

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iiiybxav -+= ( 3 ), unde i= n,1 , n>2 ,

iar necunoscutele sunt a i b

n ec. 3 avem n ec. i n+2 necunoscute , deci un sistem nedeterminat; vom

elimina nedeterminarea prin aplicarea metodei celor mai mici p trate.

[vv]=min ( 4 )

Dac introducem (3) (4)

[ ] =-+=i

iiybxavv min)( 2 ( 5 )

),(][ bafvv =

[ ]0

[ ]0

vv

a

vv

b

=

=

condi ii necesare (6)

1

1

[ ]2 ( ) 0

[ ]2 ( ) 0

n

i i

i

n

i i i

i

vva bx y

a

vv x a bx y

b

=

=

= + - =

= + - =

unde i= n,1 , ( 6 ) , ( 7 )

[ ] [ ] 0

[ ] [ ] [ ] 0

na x b y

x a xx b xy

+ - =

+ - =( 6 ) , ( 7 ) sistem normal

Calculul preciziei

Eroarea medie p tratic a unei singure m sur tori
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2

][0

-=

n

vvs

Erorile medii p tratice ale parametrilor a i b110 Qa ss = i 220 Qb ss =

Dreapta determinat cu rela ia de mai sus se nume te dreapta celor mai mici

p trate sau dreapta de regresie. Dac prelucrarea observa iilor se face

matriceal, atunci sistemul corec iilor are forma general v=Ax+l

unde

=A

n

n

x

x

x

1

1

1

2

1

)2,(KK

=

b

ax )1,2(

-

-

-

=

n

n

y

y

y

lK

2

1

)1,( ( ) +xAAT 0=lAT

=

][][

][

xxx

xnN , matricea sistemului normal

Elementele matricei inverse sunt coeficien ii :ii

Q

Dac datele experimentale se estimeaz printr-o func ie de gradul II (un arc

de parabol ), atunci vom avea :

2ii cxbxay ++= (9)

iiiiycxbxav -++= 2 (10)
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unde i= n,1 , n>3, (avem nevoie de minim 3 punte pentru curb ) ; avem o

nedeterminare, iar valoarea cea mai probabil se deduce prin metoda celor

mai mici p trate, punnd condi ia de mininim i anume :

[vv]=min (11)

Dac introducem (10) (11)

=

=-++=n

i

iii ycxbxavv1

22 min)(][ (12)

[ ]0

[ ]0

[ ]0

vv

a

vv

bvv

c

=

=

=


2

1

2

1

2 2

1

[ ]2 ( ) 0 (14)

[ ]2 ( ) 0 (15)

[ ]2 ( ) 0 (16)

n

i i i

i

n

i i i i

i

n

i i i i

i

vva bx cx y

a

vv x a bx cx y

b

vv x a bx cx y

b

=

=

=

= + + - =

= + + - =

= + + - =

Dac efectu m calculele n (14), (15), (16) ob inem:

2

2 3

2 3 4 2

[ ] [ ] [ ] 0

[ ] [ ] [ ] [ ] 0

[ ] [ ] [ ] [ ] 0

na x b x c y

x a x b x c xy

x a x b x x y

+ + - =

+ + - =

+ + - =

sistemul normal (17)

Se rezolv acest sistem apoi din (10) se calculez corec iile vi care se aplic

observa iilor yi
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Calculul preciziei


3

][0

-=

n

vvs

Erorile medii p tratice ale parametrilor a i b110 Qa ss = , 220 Qb ss = i 330 Qc ss =

Dreapta determinat cu rela ia de mai sus se nume te dreapta celor mai mici

p trate sau dreapta de regresie. Dac prelucrarea observa iilor se face

matriceal, atunci sistemul corec iilor are forma general v=Ax+l

unde

=A

2

22

21

2

1

)3,(

1

1

1

nn

n

xx

x

x

x

x

KKK

=

c

ba

x )1,3(

-

-

-

=

n

n

y

y

y

lK

2

1

)1,(

Determinarea suprafeei de tendin prin metoda celor mai

mici ptrate

Consider m un set de date experimentale (xi, yi, zi) pentru care dorim s

determin m suprafa a care descrie cel mai bine acest fenomen, adic

Z=f(x,y)
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S1 S2 S3

Sx x x

x x Sn

Fie forma general : fxyeydxcybxaz +++++= 22 (1)

Dac consider m numai termenii de ordinul 1 avem ecua ia unuio plan:

cybxaz ++= (2)

unde a-punctul n care axa y n eap planul;

b-panta n direc ia x a planului;

c-panta n direc ia y a palanului;

iiiicybxavz ++=+ (3)

iiiizcybxav -++= (4)

i= n,1 , n>3, (nr. puncte), avem un sistem nedeterminat, problema se

v-a rezolva punnd condi ia de minim.

[vv]=min (5)

Introducem (4)n (5)

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=

=-++=n

i

iii zcxbxavv1

2 min)(][ (6)

[ ]0

[ ] 0

[ ]0

vv

a

vvb

vv

c

=

=

=


1

1

1

[ ]2 ( ) 0 (7)

[ ]2 ( ) 0 (8)

[ ]2 ( ) 0 (9)

n

i i i

i

n

i i i i

i

n

i i i i

i

vva bx cy z

a

vv x a bx cy z

b

vv y a bx cy z

b

=

=

=

= + + - =

= + + - =

= + + - =

2

2

[ ] [ ] [ ] 0

[ ] [ ] [ ] [ ] 0

[ ] [ ] [ ] [ ] 0

na x b y c z

x a x b xy c zx

y a xy b y c zy

+ + - =

+ + - =

+ + - =

sistemul normal (10)

Se rezolv sistemul (10) i din (4) se determin corec iile vi

Calculul preciziilor:


3

][0

-=

n

vvs , 3 necunoscute

Erorile medii p tratice ale parametrilor a i b110 Qa ss = , 220 Qb ss = i 330 Qc ss =
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Dac prelucrareaobserva iilor se face matriceal, atunci sistemul corec iilor

are forma general v=Ax+l

unde

=A

nn yx

y

y

x

x

1

1

1

2

1

2

1

KKK

-

-

-

=

n

n

z

z

z

l

K

2

1

)1,(

ATA=N sistemul normal, iar

=

n

n

T

yyy

xxxA

....

....

1....11

21

21

Cazuri particulare:

dac n rela ia (2) consider m z=a +bx, avem ecua ia unui plan cu osingur pant , adic panta n direc ia x;

z=a+cy, avem ecua ia unui plan cu panta n direcc ia y ; z=a, avem ecua ia unui plan orizontal; n cazul vi=a-zi , sistemul normal va fi de forma:

[ ] [ ][ ] [ ] ==

==-

mediea

n

za

a

vvvvadiczna

Lcot00,0
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