Error Ellipse:

• After completing the least- square adjustment, the estimated standard deviations in coordinates of an adjusted station can be calculated from covariance matrix elements.

• These standard deviations provide error estimate in the reference axes direction.

• The graphical representation, they are half the dimension of a standard error rectangle centered at each point. The standard error rectangle has dimensions of 2Sx by 2Sy

• This is not true representation of the error present at the station

.

Y y 2S

x

2Sy

α

A

X x

B

µ

x

The equation representing a family of ellipses, centered at (µx, µy) where k2

Put x=0 and y=0 in the equation shows that the ellipse cuts the translate y axis at

±kσ

is related to the height h of the intersecting plane above the xy plane.

y (1-ρ2)1/2 and translate x axis at ±kσx (1-ρ2)

Differentiation the equation to find the gradient (dy/dx) gives the two particular

cases of tangents at y= ±kσ

1/2

y and x= ±kσx

.

The semi axes a and b are given by

a2=(1/2)(σ2x +σ2

y) + [(1/4)( σ2x -σ2

y)2 +σ2xy]

and

1/2

b2=(1/2)(σ2x +σ2

y) - [(1/4)( σ2x -σ2

y)2 +σ2xy]

1/2

f(y) f(x)

f(x,y)

h

y

x The bivariate normal distribution surface

µ

The bearing ψ of the semi major axis can be found from geometry of the ellipse and is given by

tan2ψ= 2σxy / ( σ2y -σ2

x

)

when ρ is zero (no correlation between x and y)the axes of the ellipse lie along the x and y axes.

The random variable X and Y which are in general correlated can be transformed to uncorrelated random variables by the rotation of x and y axes through the

angle ψ to coincide with the axes of the ellipse.

The transformation U = cosψ -sinψ X

V sinψ cosψ Y

If the covariance matrix for the random variable X and Y is

Cx = σ2x σ

σ

xy

yx σ2

y

The covariance matrix Cu

for the uncorrelated random variable U and V is

Cu = cosψ -sinψ σ2x σxy

sinψ cosψ σ

cosψ sinψ

yx σ2y

-sinψ cosψ

= σ2x cosψ + σ2

y sinψ -σxysin2ψ (1/2)( σ2x -σ2

y )sin 2ψ + σxy

(1/2)( σ

cos2ψ

2x -σ2

y )sin 2ψ + σxycos2ψ σ2xsin2ψ + σ2

ycos2ψ +σxy

sin2ψ

The two diagonal terms are both equal to σuv

tan 2ψ =2σ

and must be zero so

xy / (σ2y - σ2

x

).

σxy >0 σxy <0

y

x

The usual case the position of a station is uncertain in both direction and distance.

• The estimated error of the adjusted station therefore involves the errors of two jointly distribution variables (x and y coordinates).

• Thus it follows the Bivariate Normal Distribution(BND).

• The figure above shows a contour plot of the BND.

• To describe the estimated error of a station fully, it is necessary to show the orientation and lengths of the semi axes of the error ellipse.

Sy

U

Standard error rectangle

t

V

Sv

Sx

Standard error ellipse

• The orientation of the ellipse depends upon the t angle which fixes the directions of the auxiliary, orthogonal (u,v) axes along which the elipses axes lie.

• The u axis defines the weakest direction in which the station’s adjusted position is known.

• It lies in the direction not maximum expected error in the station’s coordinates.

• The v axis is orthogonal to u and defines the strongest direction in which the station’s position is known or the direction of minimum error.

• For any station, the value of t that orients the ellipse to provide these maximum and minimum values can be determined after the adjustment from the elements of the covariance matrix.