WYRÓWNANIE SIECI NIWELACYJNEJ NIEJEDNAKOWO DOKŁADNEJ
Repery nawiązania:				5, 6
Repery wyznaczane:				1, 2, 3, 4		Jacek Mentel nr 14
Niewiadome:				u = 4
Spostrzeżenia:				n = 10
Spostrzeżenia nadliczbowe:				nn = n - u = 6
Wysokości reperów nawiązania					Różnice wysokości
nr	H [m]				Lp.	od	do	DH [m]	waga (p)
5	229.823				1	4	1	2.230	1
6	231.556				2	1	5	0.501	2
					3	4	2	3.424	2
					4	2	6	1.012	4
					5	5	3	0.476	3
					6	3	6	1.258	2
					7	1	2	1.215	1
					8	1	3	0.954	2
					9	3	2	0.232	1
					10	6	1	-2.237	1
Repery wyznaczane
H1w = H10 + dh1
H2w = H20 + dh2
H3w = H30 + dh3
H4w = H40 + dh4
1. Obliczenie współrzędnych przybliżonych Hi0
H10 = H5 - L2				H10 =	229.322
H20 = H6 - L4				H20 =	230.544
H30 = H6 - L6				H30 =	230.298
H40 = H6 + L10 - L1				H40 =	227.089
2. Utworzenie układu równań poprawek
L1 + v1 = H1 - H4
v1 = H1 - H4 - L1
v1 = H1o + dh1 - H4o - dh4 - L1
v1 = dh1 - dh4 + w1
L2 + v2 = H5 - H1
v2 = H5 - H1 - L2
v2 = H5 - H1o - dh2 -L2
v2 = -dh2 + w2
L3 + v3 = H2 - H4
v3 = H2 - H4 - L3
v3 = H2o +dh2 - H4o - dh4 - L3
v3 = dh2 - dh4 + w3
L4 + v4 = H6 - H2
v4 = H6 - H2 - L4
v4 = H6 - H2o - dh2 - L4
v4 = -dh2 + w4
L5 + v5 = H3 - H5
v5 = H3 - H5 - L5
v5 = H3o + dh3 - H5 - L5
v5 = dh3 + w5
L6 + v6 = H6 - H3
v6 = H6 - H3 - L6
v6 = H6 - H3o - dh3 - L6
v6 = -dh3 + w6
L7 + v7 = H2 - H1
v7 = H2 - H1 - L7
v7 = H2o + dh2 - H1o - dh1 - L7
v7 = dh2 - dh1 + w7
L8 + v8 = H3 - H1
v8 = H3 - H1 - L8
v8 = H3o + dh3 - H1o - dh1 - L8
v8 = dh3 - dh1 + w8
L9 + v9 = H2 - H3
v9 = H2 - H3 - L9
v9 = H2o + dh2 - H3o - dh3 - L9
v9 = dh2 - dh3 + w9
L10 + v10 = H1 - H6
v10 = H1 - H6 - L10
v10 = H1o + dh1 - H6 - L10
v10 = dh1 + w10
Układ równań poprawek
v1	=	dh1			-dh4	+ w1
v2	=	-dh1				+ w2
v3	=		dh2		-dh4	+ w3
v4	=		-dh2			+ w4
v5	=			dh3		+ w5
v6	=			-dh3		+ w6
v7	=	-dh1	dh2			+ w7
v8	=	-dh1		dh3		+ w8
v9	=		dh2	-dh3		+ w9
v10	=	dh1				+ w10
Macierzowy układ równań poprawek
v = A * x + w
v	=	A				*	x	+	w
v1	=	1	0	0	-1	*		+	0.003
v2		-1	0	0	0				0.000
v3		0	1	0	-1				0.031
v4		0	-1	0	0		dh1		0.000
v5		0	0	1	0		dh2		-0.001
v6		0	0	-1	0		dh3		0.000
v7		-1	1	0	0		dh4		0.007
v8		-1	0	1	0				0.022
v9		0	1	-1	0				0.014
v10		1	0	0	0				0.003
3. Utworzenie układu równań normalnych
(AT * p * A) * x + (AT * p * w) = 0
AT										*	p										=	AT * p
										*	1	0	0	0	0	0	0	0	0	0	=
											0	2	0	0	0	0	0	0	0	0
											0	0	2	0	0	0	0	0	0	0
1	-1	0	0	0	0	-1	-1	0	1		0	0	0	4	0	0	0	0	0	0		1	-2	0	0	0	0	-1	-2	0	1
0	0	1	-1	0	0	1	0	1	0		0	0	0	0	3	0	0	0	0	0		0	0	2	-4	0	0	1	0	1	0
0	0	0	0	1	-1	0	1	-1	0		0	0	0	0	0	2	0	0	0	0		0	0	0	0	3	-2	0	2	-1	0
-1	0	-1	0	0	0	0	0	0	0		0	0	0	0	0	0	1	0	0	0		-1	0	-2	0	0	0	0	0	0	0
											0	0	0	0	0	0	0	2	0	0
											0	0	0	0	0	0	0	0	1	0
											0	0	0	0	0	0	0	0	0	1
AT * p										*	A				=	AT * p * A
										*	1	0	0	-1	=
											-1	0	0	0
											0	1	0	-1
1	-2	0	0	0	0	-1	-2	0	1		0	-1	0	0		7.000	-1.000	-2.000	-1.000
0	0	2	-4	0	0	1	0	1	0		0	0	1	0		-1.000	8.000	-1.000	-2.000
0	0	0	0	3	-2	0	2	-1	0		0	0	-1	0		-2.000	-1.000	8.000	0.000
-1	0	-2	0	0	0	0	0	0	0		-1	1	0	0		-1.000	-2.000	0.000	3.000
											-1	0	1	0
											0	1	-1	0
											1	0	0	0
AT * p										*	w	=	AT * p * w
										*	0.003	=
											0.000
											0.031
1	-2	0	0	0	0	-1	-2	0	1		0.000		-0.045
0	0	2	-4	0	0	1	0	1	0		-0.001		0.083
0	0	0	0	3	-2	0	2	-1	0		0.000		0.027
-1	0	-2	0	0	0	0	0	0	0		0.007		-0.065
											0.022
											0.014
											0.003
AT * p * A				*	x	+	AT * p * w	=	0
7.000	-1.000	-2.000	-1.000	*	dh1	+	-0.045	=	0
-1.000	8.000	-1.000	-2.000		dh2		0.083		0
-2.000	-1.000	8.000	0.000		dh3		0.027		0
-1.000	-2.000	0.000	3.000		dh4		-0.065		0
4. Rozwiązanie układu równań normalnych
(AT * p * A)-1 * (AT * p * w) = -x
(AT * p * A)-1				*	AT * p * w	=	-x
0.178	0.052	0.051	0.094	*	-0.045	=	-0.008
0.052	0.168	0.034	0.130		0.083		0.004
0.051	0.034	0.142	0.040		0.027		0.002
0.094	0.130	0.040	0.451		-0.065		-0.022
dh1 =	0.008
dh2 =	-0.004
dh3 =	-0.002
dh4 =	0.022
5. Obliczenie wyrównanych wysokości reperów
H1w = H10 + dh1 = H5 - L2 + dh1
H2w = H20 + dh2 = H6 - L4 + dh2
H3w = H30 + dh3 = H6 - L6 + dh3
H4w = H40 + dh4 = H6 + L10 - L1 + dh4
H1w =	229.330
H2w =	230.540
H3w =	230.296
H4w =	227.111
H5 =	229.823
H6 =	231.556
6. Obliczenie spostrzeżeń wyrównanych
Liw = Li + vi
v = A * x + w
A				*	x	=	A * x	+	w	=	v
1	0	0	-1	*		=	-0.013	+	0.003	=	-0.010
-1	0	0	0				-0.008		0.000		-0.008
0	1	0	-1				-0.026		0.031		0.005
0	-1	0	0		0.008		0.004		0.000		0.004
0	0	1	0		-0.004		-0.002		-0.001		-0.003
0	0	-1	0		-0.002		0.002		0.000		0.002
-1	1	0	0		0.022		-0.013		0.007		-0.006
-1	0	1	0				-0.010		0.022		0.012
0	1	-1	0				-0.002		0.014		0.012
1	0	0	0				0.008		0.003		0.011
Lw = L + v
L	+	v	=	Lw
2.230	+	-0.010	=	2.220
0.501		-0.008		0.493
3.424		0.005		3.429
1.012		0.004		1.016
0.476		-0.003		0.473
1.258		0.002		1.260
1.215		-0.006		1.209
0.954		0.012		0.966
0.232		0.012		0.244
-2.237		0.011		-2.226
7. Kontrola wyrównania
Liw = Liobl
Liobl = Hi+1 - Hi
Lp.	Hi+1	Hi	Liobl		Liw	Liobl
L1	229.330	227.111	2.220		2.220	2.220
L2	229.823	229.330	0.493		0.493	0.493
L3	230.540	227.111	3.429		3.429	3.429
L4	231.556	230.540	1.016		1.016	1.016
L5	230.296	229.823	0.473		0.473	0.473
L6	231.556	230.296	1.260		1.260	1.260
L7	230.540	229.330	1.209		1.209	1.209
L8	230.296	229.330	0.966		0.966	0.966
L9	230.540	230.296	0.244		0.244	0.244
L10	229.330	231.556	-2.226		-2.226	-2.226
8. Analiza dokładności
8a. Błąd jednostkowy
v [mm]		vv		p		pvv
-10		106		1		106
-8		71		2		142
5		27		2		53
4		17		4		67
-3		8		3		23
2		3		2		6
-6		31		1		31
12		139		2		278
12		136		1		136
11		131		1		131
					[pvv] =	974
m0 = ±	12.7	[mm]
8b. Błąd niewiadomej
Cov(x) = m02 * (AT * p * A)-1
(AT * p * A)-1				*	m02	=	Cov(x)
0.178	0.052	0.051	0.094	*		=	28.965	8.486	8.302	15.312
0.052	0.168	0.034	0.130		162.3		8.486	27.304	5.535	21.032
0.051	0.034	0.142	0.040				8.302	5.535	23.061	6.457
0.094	0.130	0.040	0.451				15.312	21.032	6.457	73.242
		= ±	5.382	[mm]
		= ±	5.225	[mm]
		= ±	4.802	[mm]
		= ±	8.558	[mm]
8c. Błąd funkcji niewiadomej
Cov(L) = A * Cov(x) * AT
A				*	Cov(x)				=	A * Cov(x)
1	0	0	-1	*					=	13.652	-12.545	1.845	-57.929
-1	0	0	0							-28.965	-8.486	-8.302	-15.312
0	1	0	-1							-6.826	6.273	-0.922	-52.210
0	-1	0	0		28.965	8.486	8.302	15.312		-8.486	-27.304	-5.535	-21.032
0	0	1	0		8.486	27.304	5.535	21.032		8.302	5.535	23.061	6.457
0	0	-1	0		8.302	5.535	23.061	6.457		-8.302	-5.535	-23.061	-6.457
-1	1	0	0		15.312	21.032	6.457	73.242		-20.478	18.818	-2.767	5.719
-1	0	1	0							-20.663	-2.952	14.759	-8.855
0	1	-1	0							0.184	21.770	-17.526	14.575
1	0	0	0							28.965	8.486	8.302	15.312
A * Cov(x)				*	AT										=	Cov(L)
13.652	-12.545	1.845	-57.929	*											=	71.581	-13.652	45.384	12.545	1.845	-1.845	-26.197	-11.807	-14.390	13.652
-28.965	-8.486	-8.302	-15.312													-13.652	28.965	6.826	8.486	-8.302	8.302	20.478	20.663	-0.184	-28.965
-6.826	6.273	-0.922	-52.210													45.384	6.826	58.483	-6.273	-0.922	0.922	13.099	5.904	7.195	-6.826
-8.486	-27.304	-5.535	-21.032		1	-1	0	0	0	0	-1	-1	0	1		12.545	8.486	-6.273	27.304	-5.535	5.535	-18.818	2.952	-21.770	-8.486
8.302	5.535	23.061	6.457		0	0	1	-1	0	0	1	0	1	0		1.845	-8.302	-0.922	-5.535	23.061	-23.061	-2.767	14.759	-17.526	8.302
-8.302	-5.535	-23.061	-6.457		0	0	0	0	1	-1	0	1	-1	0		-1.845	8.302	0.922	5.535	-23.061	23.061	2.767	-14.759	17.526	-8.302
-20.478	18.818	-2.767	5.719		-1	0	-1	0	0	0	0	0	0	0		-26.197	20.478	13.099	-18.818	-2.767	2.767	39.296	17.711	21.585	-20.478
-20.663	-2.952	14.759	-8.855													-11.807	20.663	5.904	2.952	14.759	-14.759	17.711	35.422	-17.711	-20.663
0.184	21.770	-17.526	14.575													-14.390	-0.184	7.195	-21.770	-17.526	17.526	21.585	-17.711	39.296	0.184
28.965	8.486	8.302	15.312													13.652	-28.965	-6.826	-8.486	8.302	-8.302	-20.478	-20.663	0.184	28.965
		= ±	8.461	[mm]
		= ±	5.382	[mm]
		= ±	7.647	[mm]
		= ±	5.225	[mm]
		= ±	4.802	[mm]
		= ±	4.802	[mm]
		= ±	6.269	[mm]
		= ±	5.952	[mm]
		= ±	6.269	[mm]
		= ±	5.382	[mm]
9. Zestawienia
9a. Zestawienie wysokości wyrównanych						9b. Zestawienie spostrzeżeń wyrównanych
Lp.	nr	Hw [m]	mHw [m]			Lp.	ciąg	Liw [m]	mLiw [m]
1	1	229.330	0.0054			1	4-1	2.220	0.0085
2	2	230.540	0.0052			2	1-5	0.493	0.0054
3	3	230.296	0.0048			3	4-2	3.429	0.0076
4	4	227.111	0.0086			4	2-6	1.016	0.0052
5	5	229.823	0.0000			5	5-3	0.473	0.0048
6	6	231.556	0.0000			6	3-6	1.260	0.0048
						7	1-2	1.209	0.0063
						8	1-3	0.966	0.0060
						9	3-2	0.244	0.0063
						10	6-1	-2.226	0.0054