Every fix that a navigation instrument (or a smartphone) produces is the result of a mathematical process called estimation, in which the location of the target is estimated from indirect observation. In GNSS receivers, for example, the observations are of the time of arrival of radio signals from satellites.
My new book, Location Estimation from the Ground Up, explains the models that relate the observation to the unknown location of the target, the statistical properties of these models, and the algorithms that are used to resolve the models and produce the estimates. The models and algorithms that are covered range from simple linear ones all the way to non-linear models with integer ambiguities that are used to produce centimeter-level accurate GNSS observations. The presentation combines rigor with practicality. Exercises, many with solutions, demonstrate how to develop computer code that computes fixes from time-of-arrival observations, from angle observations, from carrier-phase observations, how to estimate the time-of-arrival of GPS signals, and so on. The book also explains how to estimate the errors in fixes, both a priori and a posteriori.
I used the phrase “from the ground up” in the title because I tried hard to ensure that presentation is accessible to anyone who has gone through the first year of a degree program in engineering, math, or computer science (that is, anyone with a good grasp of the basics of linear algebra, calculus, and probability). An appendix enumerates all the mathematical knowledge that is required to follow the presentation, so if you forgot some, you know exactly what to look up. This sets the book apart from other books on GNSS and surveying techniques, which typically rely on more advanced knowledge.
The book takes a fresh and unique approach to many topics while emphasizing two recurring themes, maximum likelihood and least squares. For example, the book shows that maximum-likelihood estimation of the time of arrival of a signal leads to maximization of the cross-correlation of the received signal with a replica of the transmitted signal. Another chapter presents Kalman filtering and smoothing, showing that these problems are instances of linear least squares problems and showing how to implement the filter efficiently using a variation of a flexible algorithm from an earlier chapter.
Nicolàs de Hilster, a researcher of historic navigation and surveying instruments, has kindly allowed me to include in the book beautiful images of a theodolite and a level from his collection. These, along with fragments of two historical maps, one showing the triangulation survey of the Western United States and another from a LORAN-C navigation map, evoke an appreciation for the technological achievements of our predecessors in navigation.
I invite you to take a look at a sample chapter on the publisher’s web site (Chapter 15 on Kalman filtering and smoothing), at the table of contents and preface, or at the fairly extensive sample on Google Books. If you are interested in the mathematical modeling and in the algorithms that underlie modern navigation, I am sure you will find something useful and interesting in the book.
Location Estimation from the Ground Up, SIAM, September 2020