測量の誤差と最小二乗法

§1.1

目

次第１章

観測と誤差

観測とその種類 ······································································ 13

1.1.1.

独立観測········································································ 13

1.1.2.

条件付観測······································································ 14§1.2

誤差とその種類 ······································································ 14

1.2.1.

定誤差（系統的誤差）···························································· 14

1.2.2.

不定誤差（偶然誤差・偶差）······················································ 15

1.2.3.

過1.2.4.

不定誤差の分布と誤差の３原則···················································· 16§1.3

失·········································································· 16

最確値と最小二乗法 ································································· 17

1.3.1.

最確値と剰余（残差）···························································· 17

1.3.2.

確率曲線と最小二乗法···························································· 17

1.3.3.

精度と誤差······································································ 20§1.4

誤差の拡張（誤差の伝播） ··························································· 23

1.4.1.

誤差拡張の意義·································································· 23

1.4.2.

誤差拡張（誤差伝播）の関係式···················································· 24§1.5

算術平均（算術中数） ······························································· 24

1.5.1.

算術平均（算術中数）と剰余······················································ 24

1.5.2.

算術平均の平均二乗誤差（標準偏差）·············································· 26§1.6

一般平均（一般中数） ······························································· 32

1.6.1.

重量と一般平均·································································· 32

1.6.2.

一般平均の平均二乗誤差·························································· 33§1.7重量の拡張（重量の伝播） ··························································· 35

1.7.1.

重量の拡張の意義································································ 35

1.7.2.

重量の拡張（伝播）の関係式······················································ 35

問題Ⅰ ········································································· 43

1

第２章

観測結果の調整（その１）§2.1

観測値調整の意義···································································· 51§2.2

観測方程式による調整計算の一般式 ··················································· 51

2.2.1.

観測方程式による調整計算式（重量の等しい場合）·································· 51

2.2.2.

観測方程式による調整計算式（重量の異なる場合）·································· 54

2.2.3.

正規方程式の解法································································ 55§2.3

条件式による調整計算の一般式 ······················································· 60

2.3.1.

条件式による調整計算式（重量の等しい場合）······································ 60

2.3.2.

条件式による調整計算式（重量の異なる場合）······································ 62

2.3.3.

閉合差の重量による分配·························································· 66

2.3.4.

重量平均の意義·································································· 68§2.4

条件式による図形等の調整 ··························································· 77

2.4.1.

角条件·········································································· 77

2.4.2.

辺条件·········································································· 79

2.4.3.

辺長条件········································································ 82

2.4.4.

簡単なる図形調整における条件式の個数············································ 86

2.4.5.

条件式を求める分解法···························································· 87

2.4.6.

多角測量における簡易調整························································ 89
問題 Ⅱ ········································································ 93

第３章
§3.1

観測結果の調整（その２）及び未知量の重量と平均二乗誤差

座標調整のための観測方程式の作成 ··················································· 97

3.1.1.

座標調整のための測角に関する観測方程式·········································· 97

3.1.2.

座標調整のための測距に関する観測方程式········································· 100

3.1.3.

観測方程式の重量······························································· 102§3.2

シュライバーの法則等 ······························································ 103

3.2.1.

シュライバーの法則····························································· 103

3.2.2.

追加則········································································· 105§3.3

間接水準測量における観測方程式 ···················································· 106

3.3.1.

間接水準測量の計算式··························································· 106

3.3.2.

間接水準測量における観測方程式の作成··········································· 106§3.4

座標調整のための条件方程式の作成 ·················································· 108

3.4.1.

方向角の条件式································································· 109

3.4.2.

座標の条件式··································································· 109

2§3.5

辺長測量における条件方程式 ························································ 110§3.6

観測方程式における未知量の重量と平均二乗誤差 ····································· 112

3.6.1.

観測方程式における未知量の重量················································· 112

3.6.2.

観測方程式における未知量の平均二乗誤差········································· 115

3.6.3.

［σσ］，［ρσσ］の計算························································ 118§3.7条件方程式の未知量の重量と平均二乗誤差 ··········································· 119

3.7.1.

条件方程式の未知量の重量······················································· 119

3.7.2.

条件方程式の未知量の平均二乗誤差··············································· 121§3.8

観測方程式と条件方程式の同時回答（混合方式） ····································· 123
問題 Ⅲ ······································································· 127
総合問題 ····································································· 134

第４章

増補

§4.1

座標調整のための観測方程式の作成 ·················································· 145§4.2

観測方程式における未知量及びその関数の重量と平均二乗誤差 ························ 149

4.2.1.

観測方程式における未知量の重量················································· 149

4.2.2.

観測方程式における未知量の関数の重量··········································· 150

4.2.3.

観測方程式における未知量の平均二乗誤差とそれを求めるための［σσ］・［ρσσ］の計算 ······························· 156§4.3

条件方程式における未知量の関数の重量と平均二乗誤差 ······························· 158

4.3.1.

条件方程式における未知量の重量················································· 158

4.3.2.

条件方程式により求めた未知量の平均二乗誤差····································· 159§4.4

正規方程式の解法とその点検 ························································ 168
増補問題 ··········································································· 172
付

索録 ·········································································· 190
１

数学公式 ···································································· 190

２観測方程式関係の行列による表現 ··············································· 195３

定４ギリシヤ文字 ································································ 198

数 ···································································· 198

引3