THE ERROR ELLIPSE. 2S x. 2S x and 2S y define the dimension of the standard error rectangle. S and S show estimated error in station in direction of

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1 THE ERROR ELLIPSE Y B 2Sy 2S x A X 2S x and 2S y define the dimension of the standard error rectangle. S and S show estimated error in station in direction of x y coordinate axes. However, position of largest error at a station not necessarily aligned with either axes. PLATE 18-1

2 THE FUZZY TRAVERSE Results from error propagation in coordinates caused by distance and angle errors. PLATE 18-2

3 CONTOUR PLOT OF ADJUSTMENT SURFACE 1. Each contour depicts estimated error in unknowns at specific probability level. 2. Can increase probability level by going to lower contour. 3. Can be used in network design PLATE 18-3

4 ERROR DISTRIBUTION AT A STATION 1. At each station, there is a bivariate distribution. 2. Contour plot of distribution produces concentric ellipses, called error ellipses. 3. Probability level given by volume under distribution. 4. Higher probability produces larger error ellipse. PLATE 18-4

5 COMPONENTS OF ERROR ELLIPSE Standard Error Ellipse Sy V Y t S v Sx S u U X Standard Error Rectangle 1. t is rotation angle from Y axis to axis of largest error. 2. S is the semi-major axis of ellipse. (Largest error) u 3. S is the semi-minor axis of ellipse. (Least error) v 4. S is the standard deviation in X coordinate x 5. S is the standard deviation in Y coordinate y PLATE 18-5

6 COMPUTATION OF ELLIPSE AXIS The method for calculating the t angle, that yields the maximum and minimum semi-axes involves a twodimensional rotation. For any point I V Y x i u i u i x i Sin(t) y i Cos(t) t y i I v i U X v i x i Cos(t) y i Sin(t) or u i Sin(t) Cos(t) x i v i Cos(t) Sin(t) y i Simply Z = RX where R is the rotation matrix. PLATE 18-6

7 COMPUTATION OF ELLIPSE AXIS Problem is to develop a new covariance matrix from existing Q xx matrix which removes correlation between unknown coordinates. From G.L.O.P.O.V. Q = R Q R zz xx T q uu q uv where Q ZZ and q uv q vv q xx q xy Q xx q xy q yy Expand Q yields zz Sin 2 (t) q xx Cos(t) Sin(t) q xy Sin(t) Cos(t) q xx Cos 2 (t) q xy Sin(t) Cos(t) q xy Cos 2 (t) q yy Sin 2 (t) q xy Cos(t) Sin(t) q yy Q ZZ Cos(t) Sin(t) q xx Sin 2 (t) q xy Cos 2 (t) q xx Sin(t) Cos(t) q xy Cos 2 (t) q xy Sin(t) Cos(t) q yy Cos(t) Sin(t) q xy Sin 2 (t) q yy PLATE 18-7

8 COMPUTATION OF ELLIPSE AXIS If correlation between u and v is achieved then q will uv equal zero. Using trigonometric identities it can be shown that q uv q xx q yy 2 Sin(2t) q xy Cos(2t) 0 Solving for t yields: Tan(2t) Sin(2t) Cos(2t) 2q xy q yy q xx 2q 2t Tan 1 xy q yy q xx and q uu Sin 2 (t)q xx 2Cos(t)Sin(t)q xy Cos 2 (t)q yy q vv q xx Cos 2 (t) 2q xy Cos(t) Sin(t) q yy Sin 2 (t) PLATE 18-8

9 COMPUTATION OF ELLIPSE AXIS To compute S u and S v, do the following: S u S o q uu S v S o q vv q uu and q vv are based on rotated covariance matrix. PLATE 18-9

10 X dx Wis. dy Wis. dx Campus dy Campus Q xx EXAMPLE Assume the following: 1) S o = ± ft. 2) The covariance Q and X matrices were: xx Error Ellipse for Station Wisconsin. 2 ( ) Tan(2t) t = ' = ' Dividing by 2, t = ' Compute S u and S v: S u ± Sin 2 (t) 2( ) Cos(t)Sin(t) Cos 2 (t) ±0.25 PLATE 18-10

11 S v ± Cos 2 (t) 2( )Cos(t) Sin(t) Sin 2 (t) ±0.10 EXAMPLE When drawing ellipses, scale dimensions so that relative orientation and size of ellipses can be visualized. Ellipse Scale factor = 4800 Y Badger Wis. Y V X U Y U V X Bucky Campus PLATE 18-11

12 PROBABILITY LEVEL OF AN ERROR ELLIPSE 1. Confidence level of standard error ellipse only 39%. 2. To get higher level multiply by constant based on statistical table. a) Formula: c 2 F (, 2, degrees of freedom) b) Table of F values PLATE 18-12

13 NETWORK DESIGN BASIC PRINCIPLES: 1. Distance measurements strengthen the positions of stations in the directions collinear with the measured lines. 2. Angle and direction observations strengthen the positions of stations perpendicular to the lines of sight. 3. Largest errors occur farthest from control. 4. Adjustment can be simulated with measurements computed from approximate coordinates for stations. PLATE 18-13

14 EFFECTS OF DISTANCE AND ANGLE OBSERVATIONS WHITE SAND BUG WHITE SAND BUG SCHUTT BIRCH SCHUTT BIRCH BEAVER BEAVER BEE BEE RED BUNKER (a) Trilateration 19 Distances RED BUNKER (b) Triangulation 19 Angles Note and explain shape of ellipse at: 1. Station White, 2. Station Schutt, 3. Station Birch in figures (a) and (b). PLATE 18-14