## 1. Introduction

Each adjusted position has uncertainty based on measurement random errors. A single dimension elevation has a vertical uncertainty, ±SElev_{P}, Figure G-1.

Figure G-1 Elevation Uncertainty |

Recall that the standard deviation of a normal distribution is approximately 68% under the symmetric bell-shaped distribution curve, Figure G-2.

Figure G-2 Normal Distribution |

Standard deviation represents a 68% confidence interval. In Figure G-1, we have a 68% confidence the "true" elevation is within ±SElev_{P} of the adjusted value Elev_{P}.

A two dimension horizontal position has two uncertainties: one north, SN_{P}, one east, SE_{P,} Figure G-3.

Figure G-3 Horizontal Uncertainties |

A horizontal position has two normal distribution curves, Figure G-4, corresponding to each direction.

(a) North | (b) East |

Figure G-4 Horizontal Position Normal Curves |

Even though each direction can have a different standard deviation, the two distributions are dependent on each other. They are related by the two dimensional measurements that define the position. Instead of a single simple normal distribution, the expected positional error is a *bivariate normal distribution*. Combining both curves, Figure G-5, creates the three dimensional bivariate distribution.

Figure G-5 Bivaraiate Distribution |

The perimeter trace of horizontal "slice" through the bivariate distribution is an ellipse, Figure G-6.

Figure G-6 Error Ellipse |

There are an infinite number of error ellipses, each representing a confidence region centered on the adjusted position. The *standard error ellipse* is the defined by the standard deviations in the North and East directions.