Adjustment Computations, 6th Edition

CHAPTER 6PROPAGATION OF RANDOM ERRORS IN INDIRECTLY MEASURED QUANTITIES

6.1 BASIC ERROR PROPAGATION EQUATION

As discussed in Section 1.2, unknown values are often determined indirectly by making direct measurements of other quantities that are functionally related to the desired unknown quantities. Examples in surveying include computing station coordinates from distance and angle observations, obtaining station elevations from rod readings in differential leveling, and determining the azimuth of a line from astronomical observations. As noted in Section 1.2, since all directly observed quantities contain errors, any values computed from them will also contain errors. This intrusion, or propagation, of errors that occurs in quantities computed from direct measurements is called error propagation. This topic is one of the most important discussed in this book.

In this chapter, it is assumed that all systematic errors and mistakes have been eliminated from a set of direct observations, so that only random errors remain. To derive the basic error propagation equation, consider the simple function, z = a₁x₁ + a₂x₂, where x₁ and x₂ are two independently observed quantities with standard errors σ₁ and σ₂, and a₁ and a₂ are constants. By analyzing how errors propagate in this function, a general expression can be developed for the propagation of random errors through any function. ...