UNG DUNG PHUONG PHAP BINH SAI TRUY HOI THUAT TOAN T THUAN TRONG XU LY TOAN HOC
MANG LU'dil T R A C DIA
Lu-oTig Thanh Thach Truong Dai hpc Tai nguyen va Moi truoTig Ha Noi Tom tat
Xu- ly loan hoc (binh sai) cac mang hrai trac dia la noi dung ca ban vd quan trong cua cong tac do dac va ban do. De thuc hien cong viec nay. hien nay co nhieu phucmg phap khac nhau. Bdi bao nay de cap tai sie dimg thuat toan Tlhugn Id mot trong cac thuat loan binh sai truy hoi
Tir khoa: Binh sai truy hoi; Binli sai cac mang luai trac dia Abstract
Apply the T recurrent algorithm to the adjustment of geodetic networks Geodetic nefM'orks mathematics processing (adjustment) plays an important role in siiiyeying and mapping. A number of methods enable to solve this problem. The paper deals with the T algorithm, which belongs to recurrent adjustment methods.
Key word: Recurrent adjustment; Geodetic networks adjustment 1. Dat van de
Phuong phap binh sai truy hoi vai phep bien doi xoay Givens (thuat toan T thuan). dugc Ha MinhHoa phattrien tren nen tang ciia phuong phap binh sai truy h6i (thuat loan Q) do Markuze (1986) de xuat vai muc dich dua cac tri do vao tinh toan truy hoi dua tren co sa sir dung ma tran tam giac tren T, them vao do ma tran tam giac tren T lien he voi ma tran chuan R theo bieu thirc R^T'.T. Dac trung CO ban ciia thuat toan T thuan la tinh true tiep ma tran tam giac tren r t i r he phuong trinh so cai chinh ma khong can thong qua viec lap he phuong trinli chuan. Voi dac diem neu tren ciing voi tinh chat ciia phep bien doi xoay Givens la phep bien doi true giao nen viec tich luy sai so lam tron trong qua trinh tinh toan la nho bo qua.
Bai bao nay nghien cihi ca so ly thuyet va tien hanh chung minh bang thuc nghiem cua thuat toan T thuan de xir ly cac mang lu6i trac dia.
2. Giai quyet van de
2.1. Ly thuyet phep bien doi xoay L\' thuyet cua phep bien doi xoay dugc trinh bay trong tai lieu [4], Theo do, gia su tren mat phang Oxy khi xoay vecto a(x, y) di mot goc a, cac tga dp x.
y cua diem a se thay doi thanh (x', y') dugc tinh theo cac cong thirc:
.x'=.x.cosff - v.sina y = X s m a + y.cosa chiing ta dat ma tran:
[cos a
(1.1)
(1-2) [sin a cos a '
dugc ggi la ma tran xoay vai goc xoay a. Ma tran H la ma tran vuong goc thoa man tinh chat:
H'.H^E,^, (1.3) voi ma tran E^^ la ma tran don vi 2x2.
Liic nay, bai toan dat ra la tim goc a sao cho tung do y" cua vecto a'(x'.
y') bang 0. Phep bien doi dua a(x, y) ve a'(x', 0) dugc ggi la phep bien doi xoay.
U
Nghien cuu
Tir he phuong trinh (1.1) khi chung ta co:
.r.sinff+ v.cosa = 0
dat \ ' - 0 va thay cho (1.5) sir dung cong thirc:
(1.4) Mot trong cac nghiem ciia( 1.4) CO dang:
X y COS a =—\ sm a =
f f 6 day
/• = ^ ( x ' + / ) (1.5) Phep bien doi xoa\ trinh bay tren
dugc de xuat boi Givens (1954) de bien doi ma tran doi xirng ve dang ma tran 3 duong cheo khi giai he phuong trinh tuyen tinh voi viec xac djnh dong thoi cac gia tri rieng cua ma tran nen con dugc gpi la phep bien doi xoay Givens.
Trong thuc te ap dung phep bien doi xoay Givens, chung ta thay ma tran H a dang (1.2) bai ma tran H a dang moi nhu sau:
"C -S~\
= - ; s =
f
f
(1.7)H = (1.6)
Khi do he (1.1) CO dang moi nhu sau:
A"'= C.x-S.y
Ma tran H(\.6) thoa man tinh chdt (1.3). De ap dung phep bien doi xoay Givens trong phep bien doi ma tran doi xung xac dinh duong thanh ma tran tam giac. chiing ta bleu dien ma tran doi xung xac dinh duong R duoi dang sau:
R=R-\-^.Z' Z (1.9) 6 day ^ - ma tran doi xung xac dinh
duong. Z - vecto. /3 la mot so duang.
Doi vai ma tran doi xung xac dinli duong luon ton lai phep khai trien tam giac duy nhat theo phuong phap Cholesky.
Dieu nay co nghTa chiing ta luon co:
R = T'.T: R=T\f (1.10) a day r va r la cac ma tran tam giac tren.
Luu y (1.3) va (1.10) co the viet lai bilu thirc (1.9) dudi dang sau:
\r-o] -^'•:^.Z']
J/!-Z
[T''-.^.Z'']H''JI 4PZ
-.
B' .H'HBBieu thilc tren ggi j cho viec sit dung ma tran xoay H ii bi jn d6i hang cuoi cung ^'°' = -ipz ciia ma tran phu:
thanli hang g6m toan s6 0. Khi do chiing ta se nhan dugc ma tran phu bien doi
B = HJ"'--
vai ma tran tam giac tren T nhan dugc thoa man bieu thirc T "^ .7" = 7?.
Nhu vay. pliep bien doi Givens cho phep nhan dugc true tiep ma tran tam (l,]\) gi^c tren T thay vi phai bien doi ma tran
chuan R theo phuong phap Choleslci.
Uu diem ciia phep bien doi xoay so voi phuong phap Gauss va Cliolesliy trong viec g i a he phucmg trinh chuan la viec lian che su tich liiy cua cac sai s6 lam iron. Ngoai ra phep bien doi xoay con cho phep sir dung ky thuat ma tran thua trong qua tiinh binh sai mang lucri trie dia.
(1.12)
Tap chi Khoa hoc Tai nguyen va Moi truang - So 20 - nam 2018
2.2. Cffsff ly thuyet ciia thuat toan truy hoi T thuan
Khi ling dung thuat toan T thuan de thuc hien tinh toan, chung ta chap nhan ma tran tam giac tren ban dau co dang [4]:
T,=\0-"'E,,, (L13) 6 day m » 0 - so rat Ion (so m thuang nhan bang 6); E^^^ - ma tran don vi bac k.
Gpi R^^ - ma tran chuan dugc lap bai cac phuong trinli so cai chinh cua (i - 1) tri do dau tien. Doi voi (i - 1) tri do dau tien chung ta co he phuong trinh chuan:
R,_^.^,_,+b,_,=0 (1.14) Khi giai he phucmg trinh chuan (1.14) theo phucmg phap Choleski, chiing ta se nhan dugc 2 phuong trinh bien doi tuong duong o dang:
T,l,.iU=~b,_,-,
(1.15) a day, >_^ - vecto cac so hang tu do dugc bien doi cua he phuong trinh chuan nlian dugc sau klii dua vao tinh loan (i - 1) tri do dau lien, 7"^^- ma Iran tam giac tren nhan dugc tir phep trien Ad), = 0 , -'^,_x=[PVVl -[P^^l-y -- De thuc hien binh sai truy hoi theo thuat toan T. ngoai ma tran ban dau 7j^ dugc chgn theo cong thirc (1.13), chiing ta nhan }]^ la vecto 0, vecto hang
c""=(7^.a, M^X'r).
Khi do ma tran phu ban dau B'"' co dang:
B'" I0"".£,„ 0
khai R,_] =T^^^.T^_^ theo phuong phap Cholesky.
Khi binh sai mang luoi trac dia theo thuat loan T thuan. chung ta nhan dugc ma Iran tam giac tren T. vecto cac so hang tu do dugc bien doi Y va tong 0=[PW]. Qua trinh tinh toan truy hoi tri do thii / la _v, voi phuong trinh so hieu chinh v^=a^.dX + }i^\
trpng so P^, ma tran he so a: so hang ty do A.'"' =^X„)—_>',; vecto cac an so gan dung -X^^ dugc thuc hien theo cac nguyen tac sau:
Nguyen tac 1. Khi thuc hien phep bien doi xoay Givens vai ma tran phu:
7;_, >;_,
IF,.a, -V^.X';
chiing ta se nhan dugc ma tran phu bien doi:
~T, r, 0 , / A 0 ~ B--
B =
(1.16)
(1.17)
Bang each lan lugt dua n trj do trong mang luoi Irac dia vao tinh toan binh sai truy hoi theo nguyen tac tren
a day 7^ - ma tran tam giac tren lien he voi ma Iran chuan R, = /?,_, + a, P^a^
boi bieu thirc ^< ~ ^i -^r, vai dai lugng:
--p,.{cJ-y'-y.^y.-ry,-^ (^-i^)
chiing ta thuc hien xong qua trinh binh sai tiTjy hoi mang luoi trac dia theo phep bien doi xoay.Nguyen tac 2: Neu tri do y^ la tri do du thi CO tlie kiem tra su co mat cua cac trj do tho theo so hang ty do:
\=tj.¥,_,+>!'' (1.19) tren co so so sanh no voi gioi han:
(>-,),» =3.o-„.V^ (1.20) 6 day vecto t dugc xac dinh theo cong thuc:
TL-K^"' (1-21) Tap chi Khoa hoc Tdi nguyen vd Moi truang - So 20 - nam 20 J8 13
Nghien cuu
trpng s6 dao g^ ciia so hang tu do / dugc xac dinh theo cong thirc:
g,=P~'^t',.t, (1.22) Trpng s6 dao g^ cua so hang tu do /,
dya tren phuong phap bmh sai truy hoi CO dang:
Luu y 0_i = K-i = Vi -I'-l va b i k thirc (1.21). tir bi^u thuc tren suy ra conglhuc (1.22).
Nguyen tac w a xem xet 6 tren la CO so cua viec kiem tra su co mat cua cac trj do tho trong qua trinh binh sai truv h6i theo thuat toan 7" thuan.
De luan chirng cho co so ly thuyet bai loan tren. tien hanh tinh toan thuc nghiem mot mang luoi do cao trac dia sau [1]:
Cho luoi do cao nhu sau:
S61ieug6c: H ^ = 12.000 m S6 lieu do dugc cho a bang 1 sau:
Bang 1. Thong tin so lieu do
STT 1
1
3 4 5
Chenh cao do him) +1 M5 +5.351 +2,921 +4.853 +2 432
So tram may do n 15 30 10 10 25 3. Thirc nghiem
Hinh 1: S<r do lu&i do cao trac dia ditng tinh toan thuc nghiem Bang 2. Cac he si), trong so va so hani
3.1. Thuc nghiem binh sai truy hoi
Tinh do cao gan diing cua cac diem 1 , 2 . 3 ;
H " , - H ^ + h , - 13.935 m H ^ - H ^ + h | + h , - 19.286 m H ' \ - H ^ + h ^ = 16.853 m He so phuong trinh so cai chinh, trgng s6, s6 hang tu do dugc thong ke 6 bang 2 sau:
' tudo cua hephuffng chinh so hieu chinh
STT 1 2 3 4 5
(IH,
1 -1 -1 0 0
dH,
0 1 0 0 1
ilH,
0 0 1 1 -]
30 Trong so P, —
2 1 3 1.5 1.2
So hang tir do
i(m)'
0.000 0 000 -0.003 0.000 -0.001 Lan lugt thuc hien dua cac tri do tir 1 den 5 vao tinh toan truy hoi. moi tri do Ihyc hien biSn doi xoay k ^ 3 \kn. K^t qua cua qua trinh tinh binh sai truy h6i ciia 5 tri do dugc trinh bay o cac bang sau:
Bang 3. Ma tran phit ban ddu B'"'
B'" =
0 000001 0 0 1 414214
0 0.000001
0 0
0 0 0.000001
0
0 0 0 0
Tap chi Khoa hoc Tai nguyen va Moi truang -So 20- nam 2018
Nghien cuu Bang 4. Ma tran phu bien doi cadi cung B.
B,-
2,449490 0.000000 0.000000 0 000000
-0.408248 1.425950 0.000000 0,000000
-1,224745 -1,192188 1,666940 0,000000
-0 003674 -0,000210 0,001829 0 002304 Bang 5. Ma tran tam giac dieffc bien doi cuoi ciing T
T -
2,449490 0,000000 0,000000
-0.408248 1,425950 0,000000
-1,224745 -1,192188 1,666940 Vecto cac so hang tu do dugc bien doi
2: = (-0003674 -0.000210 0.001829f Nghiem cua bai toan dugc xac dinh theo cong thirc sau:
SX = T' I = (- 0.0008 0.0008 0.001 ij (m) Ket qua sau binli sai dugc thong ke a bang 6.
Bang 6. Ket qua sau hinh sai
Ten diem 1 2 3
Do cao gan dung (m) 13,9350 19 2860 16,8530
So hieu chinh (m) -0,0008 0 0008 0,0011
Do cao sau hinh sai (m) 13,9342 19 2868 16 8541
Do chinh xac (m) 0,0014 0,0021 0 0014 4 . K e t l u a n
So sanh ket qua sau binh sai tir bang 6 vai ket qua binh sai theo thuat loan Q dugc trinh bay trong tai lieu [6], thay rang hai phuong phap (thuat toan T thuan va thuat toan Q) cho ket qua nhu nliau.
Khi mang luoi co nhu cau phat trien mo rpng. nghla la bo sung cac diem moi. Luc nay, chiing ta hoan toan co the sir dung cac tri do moi cimg voi cac hi do cG de binh sai lai loan bp mang luoi. Tuy nhien, each lam nay se lam diay doi gia tri (tga dp, dp cao) cua cac diem khong che a giai doan Iru'oc - dieu nay khong phii hgp voi quy dinh ve do dac. Do vay, van de dat ra, hoan toan co the sir dung cac ket qua binh sai ciia mang luoi a giai doan truac dugc luu dir trong CSDL ket hgp vol cac tri do moi de lien hanh binh sai xac dinli cac diem thugc mar^ luoi ma rgng. Van de nay can tiep tuc dugc nghien cuu.
TAI LIEU THAM KHAO
[l]."Ninh Thi Kim Anh. Tran Tlii Thu Trang (2011). Giao Irinh ly ihnye! sai so. Truang Dai hoc Tai nguyen vaMoi tnrang Ha Noi, Ha Noi.
[2], Le Anh Cir6iig (2013). Nghien cihi ung dung phuang phap binh sai iniy hoi trong xii- ly so lieu tifoi ti-dc ctia. Tap chi Khoa hpc Tai nguyen va M6ilru6ng-S601,trg48- 53.
[3]. Bill Dang Quang (2012). Nghien cuu hoan thien cac phuong phap xir ly toan hoc tri do ho sung trong cac mang luoi trac dia quoc gia. Luan an tien sT ky ihuat. Truong Dai hoc Mo - Dia chat, Ha Npi.
[4]. Ha Mnih Hoa (2013). Phuong phap binh sai Iru)' hoi v&i phep bien doi xoay. Nhh xudt ban Khoa hpc v^ ky thiiSt, H& TMoi.
[5]. Bui Thj H6ng ThSm (2009). Nghen cihi dp dimg phuang phap binh sai lap de lim kiem cac tri do tho. Tap chi Khoa hpc Do dac va Ban d6. S6 l,trg. 3 7 - 4 1 ;
[6]. Luang Thanh Thach, Pham Tran Kien (2017). Ung dung phuong phap binh sai Iriiy hoi trongxu ly toan hoc frdc dia. Tap chi Khoa hpc Tai nguyen va Moi truong. So 15, trg 10 - 13,
B B T n h a n b a i : 1 2 / 3 / 2 0 1 8 , P h a n b i e n x o n g : 0 3 / 5 / 2 0 1 8
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