Compensari II - Referat 5

)0., -

COMPENSARE MSURTORILOR I STATISTIC II

PROIECT NUMRUL 5

UTCB, GEODEZIE , AN II , SERIA A , GRUPA 1

Silviu tefan BENCIU Proiect compesarea msuratorilor i statistic II

2|P a g i n a

Compensarea unui triunghi geodezic prin metoda msurtorilor de aceeai precizie, metoda msurtorilor independente de

precizii diferite i metoda msurtorilor corelate ABSTRACT: n msurtorile de precizie, pe lng valorile probabile (mediile) ale mrimilor msurate sau dedus indirect, prin calcule, ne intereseaz i precizia acestora. Asemenea problem se pune, n special, n cazul reelelor geodezice de ndesire i va fi dezbatut aici sub forma unor exemple numerice simplu.

SCOPUL LUCRRII: Este compensarea unei reele geodezice de ndesire, efectuat n figura de mai jos, folosind mai punte modele n analiza datelor brute.

CERINELE LUCRRII:

. .

. .

Tronson Baze msurate MATRICEA DE VARIANA COVARIAN

X 123.23040

1.225 10 9.402 10 7.303 10

9.402 10 1.101 10 9.485 107.303 10 9.485 10 9.373 10

Y -1235.78150 Z -749.10250

X -748.80490

5.624 10 6.673 10 5.925 10

6.673 10 6.854 10 6.589 105.925 10 6.589 10 6.833 10

Y 815.44590 Z 268.4032

X -625.58950

8.884 10 8.107 10 7.935 10

8.107 10 8.164 10 6.922 107.935 10 6.922 10 7.681 10

Y -420.32260 Z -480.71250

F C

D


3|P a g i n a

MATRICEA DE VARIANA COVARIAN:

1.225 109.402 107.303 10

000000

9.402 101.101 109.485 10

000000

7.303 109.485 109.373 10

000000

000

5.624 106.673 105.925 10

000

000

6.673 106.854 106.589 100

00

000

5.925 106.589 106.833 10

000

000000

8.884 108.107 107.935 10

000000

8.107 108.164 106.922 10

000000

7.935 106.922 107.681 10

Fiind reeaua geodezic de ndesire, realizat prin tehnologie satelitar metoda static diferenial, n care cunoatem coordonatele punctului de referin , bazele msurate , matricele de variant -covarian i abaterea standard a unitii de pondere , obinute n msurtori preliminare (brute). Se cere s se compenseze reeaua respectiv (pe coordonate relative) utiliznd modelul n urmtoarele ipoteze :

1. Msurtorile sunt independente i de aceeai precizie

2. Msurtorile sunt independente i de precizii diferite

3. Msurtorile sunt corelate

innd cont de rezultatele obinute n urma compensrii reelei n cele 3 ipoteze, s se realizeze un tabel comparativ cu indicatorii cantitativi i calitativi. Pe baza acestui tabel comparativ se vor trage concluzii referitoare la rezultatele compensrii.

Verificarea bazelor msurate pe poligoane independente (poligonul CDF) 123.23040 625.58950 748.80490 .

1235.78150 420.32260 815.44590 .

749.10250 480.7125 268.4032 .

0.01500 0.01300 0.01320 .

Calculul coordonatelor aproximative (provizorii) ale punctelor noi:

,

. . .


4|P a g i n a

. . .

Modelul funcional liniarizat sub form matriceal:

Intocmirea sistemului liniar al ecuaiilor de corecie

. . .

Matricea coeficieniilor parametrilor necunoscui sau matricea de design:

,

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 11 0 0 1 0 00 1 0 0 1 00 0 1 0 0 1

; dim 9 6

. 3 33 9 ecuaii


5|P a g i n a

3. 32 6 necunoscute

Vectorul termenilor liberi:

,

000000

000000

30123.23040 30748.80490 625.5895028764.21850 29184.55510 420.3226029250.89750 29731.59680 480.71250

,

00

0000

0.015000.013000.01320

; dim 1 9 1

Vectorul parametrilor necunoscui:

,

,

,

,

; dim 6 1

Vectorul coreciilor:

,

,

,

,

; dim 1 9 1


6|P a g i n a

Prezumia I

Msurtorile sunt independente i de aceeai precizie

Forma modelului funcional scris sub form matriceal (MODELUL DETERMINIST):

Modelul stochastic (MODELUL STATISTIC):

0

0

REGULI : ( )

0 0 2 2

0 0 2 2 0 0

,

, ,

,


7|P a g i n a

2 0 0 1 0 00 2 0 0 1 00 0 2 0 0 11 0 0 2 0 00 1 0 0 2 00 0 1 0 0 2

1 0 0 0 0 0 1 0 00 1 0 0 0 0 0 1 00 0 1 0 0 0 0 0 10 0 0 1 0 0 1 0 00 0 0 0 1 0 0 1 00 0 0 0 0 1 0 0 1

; dim 6 9

, , , , , ,

2 0 0 1 0 00 2 0 0 1 00 0 2 0 0 11 0 0 2 0 00 1 0 0 2 00 0 1 0 0 2

; dim 6 6

Proprietile matricii sistemului normal: a) este o matrice ptratic (numrul de linii este egal cu numrul de coloane n cazul acestei probleme: 6 * 6); b) este o matrice simetric fa de diagonala principal; c) Din criteriul lui Sylvester care spune c dac urmtoari determinani ai matricei (toi minorii caracteristici sunt mai mari ca zero (0))sunt mai mari ca zero, atunci matricea este pozitiv definit.

2 00 2

2 0 00 2 00 0 2

2001

020000201002


8|P a g i n a

20010

02001

0020010020

01002

2 0 0 1 0 00 2 0 0 1 00 0 2 0 0 11 0 0 2 0 00 1 0 0 2 00 0 1 0 0 2

,

, , . .

atunci:

Matricea invers se calculeaz cu relaia:

.

Matricea transpus a sistemului normal:

2 0 0 1 0 00 2 0 0 1 00 0 2 0 0 11 0 0 2 0 00 1 0 0 2 00 0 1 0 0 2

; 6 6.

Matricea adjunct a sistemului normal:


9|P a g i n a

; 6 6

unde:

18 0 0 9 0 00 18 0 0 9 00 0 18 0 0 99 0 0 18 0 00 9 0 0 18 00 0 9 0 0 18

.

cu determinantul calculat (cu metoda lui Sarrus):

, iar matricea invers devine:

1 1

18 0 0 9 0 00 18 0 0 9 00 0 18 0 0 99 0 0 18 0 00 9 0 0 18 00 0 9 0 0 18

1

0.66666667 0,00000000 0,00000000 0.33333333 0,00000000 0,000000000,00000000 0.66666667 0,00000000 0,00000000 0.33333333 0,000000000,00000000 0,00000000 0.66666667 0,00000000 0,00000000 0.333333330.33333333 0,00000000 0,00000000 0.66666667 0,00000000 0,000000000,00000000 0.33333333 0,00000000 0,00000000 0.66666667 0,000000000,00000000 0,00000000 0.33333333 0,00000000 0,00000000 0.66666667

dim 6 6.

Verificare:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


10|P a g i n a

0.66666667 0 0 0.33333333 0 0 0.33333333 0 0

0 0.66666667 0 0 0.33333333 0 0 0.33333333 0

0 0 0.66666667 0 0 0.33333333 0 0 0.33333333

0.33333333 0 0 0.66666667 0 0 0.33333333 0 0

0 0.33333333 0 0 0.66666667 0 0 0.33333333 0

0 0 0.33333333 0 0 0.66666667 0 0 0.33333333

Vectorul parametrilor necunoscui

0.005000000.004333330.004400000.005000000.00433333 0.00440000

0.00500 0.00433 0.00440 0.00500 0.00433 0.00440

Valorile cele mai probabile ale coordonatelor punctelor noi i

30123.23040 0.00500 . 28764.21850 0.00433 . 29250.89750 0.00440 .

30748.8049 0.00500 . 29184.5541 0.00433 . 29731.5968 0.00440 .

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 11 0 0 1 0 00 1 0 0 1 00 0 1 0 0 1

;

0.005000000.004333330.004400000.005000000.004333330.004400000.010000000.008666670.00880000

;

00

0000

0,01500,01300,0132


11|P a g i n a

Vectorul coreciilor

0.0050000.0043330.0044000.0050000.0043330.0044000.0050000.0043330.004400

0.00500 0.00433 0.00440 0.00500 0.00433 0.00440 0.00500 0.00433 0.00440

Calculul coordonatelor relative compensate (aplicnd coreciile coordonatelor relative msurate):

123.23040 0.00500 .

1235.78150 0.00433 . 749.10250 0.00440 .

748.80490 0.00500 . 815.44590 0.00433 . 268.40320 0.00440 .

625. 58950 0.00500 . 420.32260 0.00433 . 480.71250 0.00440 .

Verificare: Se nlocuiesc valorile coreciilor n bazele msurate pe poligoane independente (poligonul CDF):

123.22540 625.58450 748.80990 . 1235.77717 420.32693 815.45023 . 749.10690 480.70810 268.39880 .

,


12|P a g i n a

Estimarea preciziei (indicatorii preciziei): Matricial

,

0.0050000 0.00433333 0.00400000 0.00500000 0.00433333 0.00440000 0.0050000 0.00433333 0.00440000

,

,

0.0018941

atunci: 1.) Conform teoriei generale n cazul Gauss-Markov, abaterea standard de selecie a unei msurtori

se poate calcula cu relaia (estimatea abaterii standard):

0.00189419 6

0.00189413

0.013762751.73205080

0.00795

n care n reprezint numrul msurtorilor (n = 9) i h numrul necunoscutelor implicate n model (2 (dou) puncte noi 3 = 6 = h). 2.) Abaterea standard (eroarea medie) a unei necunoscute estimarea parametrilor empirici:

0.00795 0.66666667 0.00649

0.00795 0.66666667 0.00649

0.00795 0.66666667 0.00649

0.00795 0.66666667 0.00649

0.00795 0.66666667 0.00649

0.00795 0.66666667 0.00649

unde reprezint coeficienii de pondere ptratici din matricea invers (mat. cofactorilor) , .

Indicator Cantitativ[m] Indicator

Calitativ[m]

30123.22540 0.00649 28764.22283 0.00649 29250.89310 0.00649

30748.80990 0.00649 29184.54977 0.00649 29731.60120 0.00649


13|P a g i n a

Prezumia a II - a

Msurtorile sunt independente i de precizii diferite



0

0

REGULI : ( )

0 0 2 2

0 0 2 2 0 0

,

, ,

,


14|P a g i n a

1.651357 0 0 0.200110 0 00 1.832452 0 0 0.217758 00 0 2.128152 0 0 0.231451

0.200110 0 0 0.516216 0 00 0.217758 0 0 0.477136 00 0 0.231451 0 0 0.491627

1.225 10 0 0 0 0 0 0 0 00 1.101 10 0 0 0 0 0 0 00 0 9.373 10 0 0 0 0 0 00 0 0 5.624 10 0 0 0 0 00 0 0 0 6.854 10 0 0 0 00 0 0 0 0 6.833 10 0 0 00 0 0 0 0 0 8.884 10 0 00 0 0 0 0 0 0 8.164 10 00 0 0 0 0 0 0 0 7.681 10

81632.653061 0 0 0 0 0 0 0 00 90826.521344 0 0 0 0 0 0 00 0 106689.427078 0 0 0 0 0 00 0 0 17780.938834 0 0 0 0 00 0 0 0 14590.020426 0 0 0 00 0 0 0 0 14634.860237 0 0 00 0 0 0 0 0 11256.190905 0 00 0 0 0 0 0 0 12248.897599 00 0 0 0 0 0 0 0 13019.138133

Verificare :

1 .


15|P a g i n a

1.451247 0 0 0 0 0 0 0 00 1.614694 0 0 0 0 0 0 00 0 1.896701 0 0 0 0 0 00 0 0 0.316106 0 0 0 0 00 0 0 0 0.259378 0 0 0 00 0 0 0 0 0.260175 0 0 00 0 0 0 0 0 0.200110 0 00 0 0 0 0 0 0 0.217758 00 0 0 0 0 0 0 0 0.231451

1 0 0 0 0 0 1 0 00 1 0 0 0 0 0 1 00 0 1 0 0 0 0 0 10 0 0 1 0 0 1 0 00 0 0 0 1 0 0 1 00 0 0 0 0 1 0 0 1

; dim 6 9

1.451247 0 0 0 0 0 0.200110 0 00 1.614694 0 0 0 0 0 0.217758 00 0 1.896701 0 0 0 0 0 0.2314510 0 0 0.316106 0 0 0.200110 0 00 0 0 0 0.259378 0 0 0.217758 00 0 0 0 0 0.260175 0 0 0.231451

Matricea sistemului normal: , , , ,

1.651357 0 0 0.200110 0 00 1.832452 0 0 0.217758 00 0 2.128152 0 0 0.231451

0.200110 0 0 0.516216 0 00 0.217758 0 0 0.477136 00 0 0.231451 0 0 0.491627

; dim 6 6

Proprietile matricii sistemului normal: a) este o matrice ptratic (numrul de linii este egal cu numrul de coloane n cazul acestei probleme: 6 * 6);


16|P a g i n a

b) este o matrice simetric fa de diagonala principal; c) Din criteriul lui Sylvester care spune c dac urmtoari determinani ai matricei (toi minorii caracteristici sunt mai mari ca zero (0))sunt mai mari ca zero, atunci matricea este pozitiv definit. 1.651357

1.651357 00 1.832452 .

1.651357 0 0

0 1.832452 00 0 2.128152

.

1.651357

00

0.200110

01.832452

00

00

2.1281520

0.200110

00

0.516216 .

1.651357

00

0.2001100

0

1.83245200

0.217758

00

2.12815200

0.200110

00

0.5162160

0

0.21775800

0.477136 .

1.651357 0 0 0.200110 0 0

0 1.832452 0 0 0.217758 00 0 2.128152 0 0 0.231451

0.200110 0 0 0.516216 0 00 0.217758 0 0 0.477136 00 0 0.231451 0 0 0.491627

.

, ,

, , , , . .

atunci:



17|P a g i n a

.


1.651357 0 0 0.200110 0 0

0 1.832452 0 0 0.217758 00 0 2.128152 0 0 0.231451

0.200110 0 0 0.516216 0 00 0.217758 0 0 0.477136 00 0 0.231451 0 0 0.491627

; 6 6.


; 6 6

unde:

1.101256 0 0 0.133449 0 00 1.222025 0 0 0.145218 00 0 1.419221 0 0 0.154350

0.133449 0 0 0.344254 0 00 0.145218 0 0 0.318192 00 0 0.154350 0 0 0.327856

.


. , iar matricea invers devine:

1 1.

1.101256 0 0 0.133449 0 00 1.222025 0 0 0.145218 00 0 1.419221 0 0 0.154350

0.133449 0 0 0.344254 0 00 0.145218 0 0 0.318192 00 0 0.154350 0 0 0.327856


18|P a g i n a

1

0.635411 0,000000 0,000000 0.246316 0,000000 0,0000000,000000 0.577011 0,000000 0,000000 0.263339 0,0000000,000000 0,000000 0.495249 0,000000 0,000000 0.2331570.246316 0,000000 0,000000 2.032659 0,000000 0,0000000,000000 0.263339 0,000000 0,000000 2.216021 0,0000000,000000 0,000000 0.233157 0,000000 0,000000 2.143831

dim 6 6.

Verificare:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.635411 0 0 0.246319 0 0 0.389095 0 00 0.577011 0 0 0.263339 0 0 0.313671 00 0 0.495249 0 0 0.233157 0 0 0.262092

0.246316 0 0 2.032659 0 0 1.786343 0 00 0.263339 0 0 2.216021 0 0 1.952682 00 0 0.233157 0 0 2.143831 0 0 1.910674

0.922138 0 0 0.077862 0 0 0.077862 0 00 0.931696 0 0 0.060662 0 0 0.068304 00 0 0.939338 0 0 0.060505 0 0 0.060662

0.357465 0 0 0.642535 0 0 0.357465 0 00 0.425212 0 0 0.574788 0 0 0.425212 00 0 0.442228 0 0 0.557772 0 0 0.442228


0.0011680.0008880.0008010.0053620.0055280.005837

0.00117 0.00089 0.00080 0.00536 0.00553 0.00584


19|P a g i n a


30123.23040 0.00117 . 28764.21850 0.00089 . 29250.89750 0.00080 .

30748.8049 0.00536 . 29184.5541 0.00553 . 29731.5968 0.00584 .

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 11 0 0 1 0 00 1 0 0 1 00 0 1 0 0 1

;

0.001167930.000887960.000800730.005361980.005527760.005837410.006529910.006415720.00663814

;

00

0000

0,015000,013000,01320

Vectorul coreciilor

0.0011680.0008880.0008010.0053620.0055280.0058370.0084700.0065840.006562

0.00117 0.00089 0.00080 0.00536 0.00553 0.00584 0.00847 0.00658 0.00656



20|P a g i n a

123.23040 0.00117 .

1235.78150 0.00089 . 749.10250 0.00080 .

748.80490 0.00536 . 815.44590 0.00553 . 268.40320 0.00584 .

625. 58950 0.00847 . 420.32260 0.00658 . 480.71250 0.00656 .


123.22923 625.58103 748.81026 , 1235.78061 420.32918 815.45143 , 749.10330 480.70594 268.39736 ,

,


,

0.00116793 0.00088796 0.00080073 0.00536198 0.00552776 0.00583741 0.00847009 0.00658428 0.00656186

0.00169494 0.00143378 0.00151875 0.00169495 0.00143378 0.00151875 0.00169495 0.001433784 0.00151875 0,00006411

atunci: 1.) Conform teoriei generale n cazul Gauss-Markov, abaterea standard de selecie a unitii

depondere se poate calcula cu relaia (estimatea abaterii standard):

0.000064119 6

0.000064113

0.008006931.73205080

0.00462


21|P a g i n a


0.00462 0.63541084 0.00368

0.00462 0.57701068 0.00351

0.00462 0.49524858 0.00325

0.00462 2.03265884 0.00659

0.00462 2.21602143 0.00688

0.00462 2.14383092 0.00677



Calitativ[m]

30123.22923 0.00368 28764.21939 0.00351 29250.89670 0.00325

30748.81026 0.00659 29184.54857 0.00688 29731.60264 0.00677


22|P a g i n a

Prezumia a III - a

Msurtorile sunt corelate



0

0

REGULI : ( )

0 0 2 2

0

0 2 2 0 0

,

, ,

,


23|P a g i n a

1.651555 0.014268 0.009133 0.200146 0.001970 0.0020500.014268 1.832733 0.014304 0.001970 0.217794 0.0019420.009133 0.014304 2.128442 0.002050 0.001942 0.2314900.200146 0.001970 0.002050 0.516317 0.005022 0.0006620.001970 0.217794 0.001942 0.005022 0.477226 0.0005330.002050 0.001942 0.231490 0.000662 0.000533 0.491713

1.225 105 9.402 108 7.303 108 0 0 0 0 0 09.402 108 1.097 105 9.485 108 0 0 0 0 0 07.303 108 9.485 108 9.373 106 0 0 0 0 0 0

0 0 0 5.624 105 6.673 107 5.925 107 0 0 00 0 0 6.673 107 6.854 105 6.589 107 0 0 00 0 0 5.925 107 6.589 107 6.833 105 0 0 00 0 0 0 0 0 8.884 105 8.107 107 7.935 107

0 0 0 0 0 0 8.107 107 8.164 105 6.922 1070 0 0 0 0 0 7.935 107 6.922 107 7.681 105

81641.712937 691.760439 629.113498 0 0 0 0 0 0691.760439 90840.301515 913.867847 0 0 0 0 0 0629.113498 913.867847 106703.576712 0 0 0 0 0 0

0 0 0 17784.583110 171.682700 152.557351 0 0 00 0 0 171.682700 14593.030390 139.230583 0 0 00 0 0 152.557351 139.230583 14637.525673 0 0 00 0 0 0 0 0 11258.232063 110.818637 115.3065810 0 0 0 0 0 110.818637 12250.924418 109.2586290 0 0 0 0 0 115.306581 109.258629 13021.313951

Verificare :

1 .


24|P a g i n a

1.451408 0.012298 0.011184 0 0 0 0 0 00.012298 1.614939 0.016247 0 0 0 0 0 00.011184 0.016247 1.896952 0 0 0 0 0 0

0 0 0 0.316170 0.003052 0.002712 0 0 00 0 0 0.003052 0.259432 0.002475 0 0 00 0 0 0.002712 0.002475 0.260223 0 0 00 0 0 0 0 0 0.201046 0.001970 0.0020500 0 0 0 0 0 0.001970 0.217794 0.0019420 0 0 0 0 0 0.002050 0.001942 0.231490

1 0 0 0 0 0 1 0 00 1 0 0 0 0 0 1 00 0 1 0 0 0 0 0 10 0 0 1 0 0 1 0 00 0 0 0 1 0 0 1 00 0 0 0 0 1 0 0 1

; dim 6 9

1

1.451408 0.012298 0.011184 0 0 0 0.200146 0.001970 0.0020500.012298 1.614939 0.016247 0 0 0 0.001970 0.217794 0.0019420.011184 0.016247 1.896952 0 0 0 0.002050 0.001942 0.231490

0 0 0 0.316170 0.003052 0.002712 0.200146 0.001970 0.0020500 0 0 0.003052 0.259432 0.002475 0.001970 0.217794 0.0019420 0 0 0.002712 0.002475 0.260223 0.002050 0.001942 0.231490

Matricea sistemului normal: , ,

,

,

1.651555 0.014268 0.009134 0.200146 0.001970 0.0020500.014268 1.832733 0.014304 0.001970 0.217794 0.0019420.009134 0.014304 2.128443 0.002050 0.001942 0.2314900.200146 0.001970 0.002050 0.516317 0.005022 0.0006620.001970 0.217794 0.001942 0.005022 0.477226 0.0005330.002050 0.001942 0.231490 0.000662 0.000533 0.491713

; dim 6 6

Proprietile matricii sistemului normal: a) este o matrice ptratic (numrul de linii este egal cu numrul de coloane n cazul acestei probleme: 6 * 6);


25|P a g i n a

b) este o matrice simetric fa de diagonala principal; c) Din criteriul lui Sylvester care spune c dac urmtoari determinani ai matricei (toi minorii caracteristici sunt mai mari ca zero (0))sunt mai mari ca zero, atunci matricea este pozitiv definit. 1.651555

1.651555 0.0142680.014268 1.832452 .

1.651555 0.014268 0.0091340.014268 1.832452 0.0143040.009134 0.014304 2.128152

.

1.6515550.0142680.0091340.200146

0.0142681.8324520.0143040.001970

0.0091340.0143042.1281520.002050

0.2001460.0019700.0020500.516317

.

1.6515550.0142680.0091340.2001460.001970

0.0142681.8324520.0143040.0019700.217794

0.0091340.0143042.1281520.0020500.001942

0.2001460.0019700.0020500.5163170.005022

0.0019700.2177940.0019420.0050220.477226

.

1.651555 0.014268 0.009134 0.200146 0.001970 0.0020500.014268 1.832733 0.014304 0.001970 0.217794 0.0019420.009134 0.014304 2.128443 0.002050 0.001942 0.2314900.200146 0.001970 0.002050 0.516317 0.005022 0.0006620.001970 0.217794 0.001942 0.005022 0.477226 0.0005330.002050 0.001942 0.231490 0.000662 0.000533 0.491713

.

, ,

, , , , . .

atunci:



26|P a g i n a

.


1.651555 0.014268 0.009134 0.200146 0.001970 0.0020500.014268 1.832733 0.014304 0.001970 0.217794 0.0019420.009134 0.014304 2.128443 0.002050 0.001942 0.2314900.200146 0.001970 0.002050 0.516317 0.005022 0.0006620.001970 0.217794 0.001942 0.005022 0.477226 0.0005330.002050 0.001942 0.231490 0.000662 0.000533 0.491713

; 6 6.


; 6 6

unde:

1.102110 0.009521 0.006095 0.133561 0.001315 0.0013680.009521 1.223013 0.009545 0.001315 0.145338 0.0012960.006095 0.009545 1.420345 0.001368 0.001296 0.1544770.133561 0.001315 0.001368 0.344547 0.003351 0.0004420.001315 0.145338 0.001296 0.003351 0.318461 0.0003560.001368 0.001296 0.154477 0.000442 0.000356 0.328128

.


. , iar matricea invers devine:

1 1.

1.102110 0.009521 0.006095 0.133561 0.001315 0.0013680.009521 1.223013 0.009545 0.001315 0.145338 0.0012960.006095 0.009545 1.420345 0.001368 0.001296 0.1544770.133561 0.001315 0.001368 0.344547 0.003351 0.0004420.001315 0.145338 0.001296 0.003351 0.318461 0.0003560.001368 0.001296 0.154477 0.000442 0.000356 0.328128


27|P a g i n a

1

0.635408 0.004983 0.003506 0.246334 0.002263 0.0046530.004983 0.577008 0.004662 0.002316 0.263360 0.0047830.003506 0.004662 0.495243 0.003650 0.004428 0.2331950.246334 0.002316 0.003650 2.032507 0.021451 0.0055150.002263 0.263360 0.004428 0.021451 2.215876 0.0055640.004653 0.004783 0.233195 0.005515 0.005564 2.143544

dim 6 6.

Verificare:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.635408 0.004983 0.003506 0.246334 0.002263 0.004653 0.389074 0.002720 0.0011480.004983 0.577008 0.004662 0.002316 0.263360 0.004783 0.002667 0.313647 0.0001210.003506 0.004662 0.495243 0.003650 0.004428 0.233195 0.000144 0.000234 0.2620480.246334 0.002316 0.004650 2.032507 0.021451 0.005515 1.786173 0.019135 0.0018650.002263 0.263360 0.004428 0.021451 2.215876 0.005564 0.019188 1.952515 0.0011370.004653 0.004783 0.233195 0.005515 0.005564 2.143544 0.000862 0.000781 1.910349

1

0.922136 0.000176 0.000537 0.077864 0.000176 0.000537 0.077864 0.000176 0.0005370.000084 0.931695 0.000587 0.000084 0.068305 0.000587 0.000084 0.068305 0.0005870.000508 0.000560 0.939338 0.000508 0.000560 0.060662 0.000508 0.000560 0.0606620.357462 0.000652 0.004130 0.642538 0.000652 0.004130 0.357462 0.000652 0.0041300.000004 0.425211 0.004095 0.000004 0.574789 0.004095 0.000004 0.425211 0.0040950.004087 0.003879 0.442230 0.004087 0.003879 0.557770 0.004087 0.003879 0.442230


1

0.0011730.0008940.0008160.0054250.0055820.005949

0.00117 0.00089 0.00082 0.00542 0.00558 0.00595


28|P a g i n a


30123.23040 0.00117 . 28764.21850 0.00089 . 29250.89750 0.00082 .

30748.8049 0.00542 . 29184.5541 0.00558 . 29731.5968 0.00595 .

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 11 0 0 1 0 00 1 0 0 1 00 0 1 0 0 1

;

0.001172760.000894440.000815640.005424930.005581740.005949170.006597690.006476180.00676481

;

00

0000

0,015000,013000,01320

Vectorul coreciilor

0.0011730.0008940.0008160.0054250.0055820.0059490.0084020.0065240.006435

0.00117 0.00089 0.00082 0.00542 0.00558 0.00595 0.00840 0.00652 0.00644



29|P a g i n a

123.23040 0.00117 .

1235.78150 0.00089 . 749.10250 0.00082 .

748.80490 0.00542 . 815.44590 0.00558 . 268.40320 0.00595 .

625. 58950 0.00840 . 420.32260 0.00652 . 480.71250 0.00644 .


123.22923 625.58110 748.81032 , 1235.78061 420.32912 815.45148 , 749.10332 480.70606 268.39725 ,

,


,

0.00117276 0.00089444 0.00081564 0.00542493 0.00558174 0.00594917 0.00840231 0.00652382 0.00643519

1 0.00168203 0.00141680 0.00151958 0.00168203 0.00141680 0.00151958 0.00168203 0.00141680 0.00151958

0.00006371atunci: 1.) Conform teoriei generale n cazul Gauss-Markov, abaterea standard de selecie a unitii

depondere se poate calcula cu relaia (estimatea abaterii standard):

0.00006371

3 0.007981681.73205080

0.00461


30|P a g i n a


0.00461 0.63540805 0.00367

0.00461 0.57700762 0.00350

0.00461 0.49524340 0.00324

0.00462 2.03250669 0.00657

0.00462 2.21587568 0.00686

0.00462 2.14354387 0.00675



Calitativ[m]

30123.22923 0.00367 28764.21939 0.00350 29250.89668 0.00324

30748.81032 0.00657 29184.54852 0.00686 29731.60275 0.00675

Silviu tefan BENC

IU

Proiect com

pesarea msuratorilor i statistic II

Tabel comparativ cu rezultatele compensrii n cele 3 (trei) ipoteze

Msurtori independente de aceeai precizie

Msurtori independente de precizii diferite

Msurtori corelate

Indicator cantitativ Abaterea standard de selecie a unei msurtori

Indicator cantitativ Abaterea standard de selecie a unitii de pondere

Indicator cantitativ Abaterea standard de selecie a unitii de pondere

0.00795

0.00462

0.00461

Indicator calitativ Estimarea preciziei

parametrilor necunoscui Indicator calitativ Estimarea preciziei

parametrilor necunoscui Indicator calitativ Estimarea preciziei

parametrilor necunoscui

0.00649

0.00649

0.00649

0.00649

0.00649

0.00649

0.00368

0.00351

0.00325

0.00659

0.00688

0.00677

0.00367

0.00350

0.00324

0.00657

0.00686

0.00675

Din Tabel comparativ, n care se prezint valorile obinute prin compensare n cele 3 (trei) prezumii se poate observa c indiferent de ipoteza avut n vedere: msurtori independente de aceeai precizie, msurtori independente de precizii diferite, respectiv msurtori corelate, dac avem n vedere indicatorii calitativi, se estimeaz precizii ale parametrilor necunoscui care variaz foarte puin de la o prezumie la cealalt.

31|P

agin

a


Dup compensarea reelei geodezice de ndesire prin cele 3 prezumii, putem observa c ipotezele II i III, care presupun c msurtorile sunt de precizii diferite (ponderate), compeseaz cel mai bine reeaua respectiv.

Din punct de vedere cantitativ, cele 3 prezumii, nu prezint mari diferene n determinarea valorilor cele mai probabile ale coordonatelor punctelor noi, ns putem observa din nou efectul msurtorilor ponderate (ncrederea pe care o acord acestora), prezumiile II i III determinnd aproximativ aceleai valori ale punctelor noi.

Din punct de vedere calitativ, prezumia I (prin definiie) ofer aceeai ncredere tuturor msurtorilor. prezumiile II i III determin valorile cele mai probabile ale coordonatelor punctelor noi cu o precizie mai bun, observndu-se iar asemnarea dintre aceste dou ipoteze, datorat efectului ponderii care influeneaz reeaua respectiv.

Dintre toate ipotezele, putem spune c acurateea cea mai mare n compensarea reelei geodezice de ndesire o ofer prezumia III, datorit dependenei fizice dintre mrimile msurate de care ine cont.

X Y Z

[ m ] DIAGRAMA COMPARATIV D [X, Y, Z]

Prezumtia1Prezumtia2Prezumtia3

32|P a g i n a


33|P a g i n a

Interpretarea privind elipsa erorilor i elipsoidul erorilor Cu erorile (abaterile) i se poate determina elipsa erorilor, care este un domeniu de ncredere (plan), astfel:

2

Dac se cere abaterea standard pe o direcie oarecare, se folosete elipsa erorilor n spaiul cu dou dimensiuni (2-D), respentiv elipsaodul erorilor, n spaiul cu trei dimensiuni (3-D). Se mai poate determina i eroarea total (sau eroarea Helmert) astfel:

unde:

i reprezint erorile medii ale necunoscutelor.

X Y Z

[ m ] DIAGRAMA COMPARATIV F [X, Y, Z]

Prezumtia1Prezumtia2Prezumtia3


34|P a g i n a

Elipsa erorilor reprezentare practic general.

CLASIFICAREA REELELOR GEODEZICE (n funcie de elipsa erorilor)

1.) Reele omogene: acele reele pentru care elipsele erorilor au aceeai form, orientare i dimensiune n punctele noi.

2.) Reele izotrope: acele reele n cadrul crora eroarea pe orice direcie ntr-un punct este aceei (atunci elipsa poate deveni cerc). Domeniu de studiu n anul 4 la msurtori GNSS (Global Navigation Satellite System).

3.) Cazul ideal: cnd avea de-a face cu ambele tipuri de reele (omogene i izotrope), atunci cnd elipsele devin cercuri de raze egale.


35|P a g i n a

Dac avem o reea geodezic format din p - puncte noi , atunci elementele matricei de covarian se calculeaz cu ajutorul blocurilor bidimensionale dup diagonala principal a matricei inverse a sistemului normal .

Interpretarea statistic Din punctul de vedere al indicatorilor cantitativi, prin prisma rezultatelor obinute, de 0.00795 m (n ipoteza msurtorilor independente de aceeai precizie), 0.00462 (n ipoteza msurtorilor independente de precizii diferite), respectiv 0.00461 (n ipoteza msurtorilor corelate) i avnd n vedere faptul c dispersia (abaterea standard) msoar gradul de mprtiere al variabilei fa de medie (abaterea standard tinde ctre dispersie ntre 30 de msurtori directe), se poate considera evoluia acestor indicatori ca respectnd legea distrbuiei normale, cu valori ale probabilitilor de 68%, 95%, respectiv 99%.

Aceste valori arat c probabilitile ca o variabil aleatoare X , s nu devieze de la media cu mai mult de , 2 , respectiv 3 sunt respectiv 68%, 95%, respectiv 99% n funcie de acurateea (precizia) presupus, deci p = 0,95 (vrem s fim 95% siguri de corectitudinea concluziei rezultate n urma testului). n acest caz vom folosi un nivel de semnificatie " risc " care se calculeaz n modul devenit deja cunoscut:

, %

Pr5 _sem2Pr5 _sem2

Documents

Compensari II - Referat 5