LEAST-SQUARES ADJUSTMENTS - GPS Satellite Surveying - page 150

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which yields the y 1 coordinate for the point of tangency:
y 1 =σ 0 √ q x 1 ρ x 1 ,x 2 =σ x 1 ρ x 1 ,x 2 (4.326)

The equation for the horizontal tangent is
y 1 =σ 0 √ q x 1 (4.327)

It follows from the numerator of (4.323) that
y 2 =σ 0 √ q x 2 ρ x 1 ,x 2 =σ x 2 ρ x 1 ,x 2

(4.328)

Figure 4.9 shows that the standard ellipse becomes narrower the higher the correla-
tion. For correlation plus or minus 1 (linear dependence), the ellipse degenerates into
the diagonal of the rectangle. The ellipse becomes a circle if a

[15

= σ = σ

b ,or and

x 1 x 2

ρ x 1 ,x 2 = 0.

Lin
—
0.0
——
Nor
*PgE

4. 9.7 Other Measures of Precision

In surveying and geodesy, the most popular measure of precision is the standard de-
viation. The confidence regions are usually expressed in terms of ellipses and ellip-
soids of standard deviation. These figures are often scaled to contain 95% probability
or higher. Because GPS is a popular tool for both surveying and navigation, several
of the measures of precision used in navigation are becoming increasingly popular in
surveying. Examples include the dilution of precision (DOP) numbers. The DOPs are
discussed in detail in Section 7.4.1. Other single-number measures refer to circular
or spherical confidence regions for which the eigenvalues of the cofactor matrix have
the same magnitude. In these cases, the standard deviations of the coordinates and
the semiaxes are of the same size. See Equation (4.297). When the standard devia-
tions are not equal, these measures become a function of the ratio of the semiaxes.
The derivation of the following measures and additional interpretation are given in
Greenwalt and Shultz (1962).
The radius of a circle that contains 50% probability is called the circular error
probable (CEP). This function is usually approximated by segments of straight lines.
The expression

[15

0 . 5887 σ

x 2

= +σ

CEP (4.329)

x 1

is, strictly speaking, valid in the region σ min / σ max ≥ 0 . 2, but it is the function used

m ost often. The 90% probability region

CMAS = 1 . 8227 × CEP (4.330)

is called the circular map accuracy standard. The mean position error (4.319) is also
called the mean square positional error (MSPE), or the distance root mean square
(DRMS), i.e.,