# CHAPTER 13PRECISIONS OF INDIRECTLY DETERMINED QUANTITIES

## 13.1 INTRODUCTION

Following an adjustment, it is important to know the estimated errors in both the derived quantities and the adjusted observations. For example, after adjusting a level net as described in Chapter 12, the uncertainties in both computed bench mark elevations and adjusted elevation differences can be determined. In Chapter 5, error propagation formulas were developed for indirectly measured quantities that were functionally related to observed values. In this chapter, error propagation formulas are developed for the quantities computed in a least squares solution.

## 13.2 DEVELOPMENT OF THE COVARIANCE MATRIX

Consider an adjustment involving weighted observation equations like those in the level circuit example of Section 12.4. The matrix form for the system of weighted observation equation is

and the least squares solution of the weighted observation equations is given by

In this equation, *X* contains the most probable values for the unknowns, whereas the true values are *X _{true}*. The true values differ from

*X*by some small amount Δ

*X*, such that

where Δ*X* represents the true errors in ...

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