F. Nonlinear Measurements

1. Introduction

Recall that non-linear equations solution requires an iterative approach after applying the Taylor Series. In an LS application, this means initial approximations are used to formulate the [C] and [K] matrices and the [U] solution matrix has corrections for the approximations.The [C] and [K] matrices are updated and a new set of corrections determined. This repetitive process continues until the corrections are small enough to be acceptable.

The elevation observation equation, being single dimension, is linear so a leveling circuit does not require an iterative solution. A horizontal network uses distance and angles to determine two dimensional point positions. The relationship between measurements and positions is based on higher math including trigonometric functions which are not linear.

A horizontal network can be adjusted by coordinate variation. Beginning with a mathematical equation relating a measurement type to coordinates, a Taylor Series is applied to create an observation equation. Initial coordinate approximations of the unknown points are used with the observation equations for the LS adjustment to generate coordinate corrections. These are applied and the adjustment repeated until the corrections fall below a threshold. The final results are coordinates which can be used, if desired, to solve adjusted distances, azimuths, and angles.

This chapter covers three primary horizontal measurement types: distance, azimuth, and angle. Observation equations are developed to help understand the iterative nature of a horizontal adjustment. An example simple network shows how the observation equations are used in the adjustment process. Be forewarned: manually adjusting a horizontal network involves a lot of computations. Not only can it be tedious, but math errors can be easily made. 

What you should get out of this chapter is that there is a lot of math involved (including calculus) in a simple horizontal network adjustment. Increasing the degrees of freedom substantially impacts the calculations which must be performed. Understanding why an iterative appraoch is necessary to adjuist the network is equally important.