# A Minimum Variance Algorithm for Overdetermined TOA Equations with an Altitude Constraint.

## Abstract

We present a direct (non-iterative) method for solving for the location of a radio frequency (RF) emitter, or an RF navigation receiver, using four or more time of arrival (TOA) measurements and an assumed altitude above an ellipsoidal earth. Both the emitter tracking problem and the navigation application are governed by the same equations, but with slightly different interpreta- tions of several variables. We treat the assumed altitude as a soft constraint, with a specified noise level, just as the TOA measurements are handled, with their respective noise levels. With 4 or more TOA measurements and the assumed altitude, the problem is overdetermined and is solved in the weighted least squares sense for the 4 unknowns, the 3-dimensional position and time. We call the new technique the TAQMV (TOA Altitude Quartic Minimum Variance) algorithm, and it achieves the minimum possible error variance for given levels of TOA and altitude estimate noise. The method algebraically produces four solutions, the least-squares solution, and potentially three other low residual solutions, if they exist. In the lightly overdermined cases where multiple local minima in the residual error surface are more likely to occur, this algebraic approach can produce all of the minima even whenmore »

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1436065

- Report Number(s):
- SAND-2018-4185

662700

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Romero, Louis A, and Mason, John J.
```*A Minimum Variance Algorithm for Overdetermined TOA Equations with an Altitude Constraint.*. United States: N. p., 2018.
Web. doi:10.2172/1436065.

```
Romero, Louis A, & Mason, John J.
```*A Minimum Variance Algorithm for Overdetermined TOA Equations with an Altitude Constraint.*. United States. doi:10.2172/1436065.

```
Romero, Louis A, and Mason, John J. Sun .
"A Minimum Variance Algorithm for Overdetermined TOA Equations with an Altitude Constraint.". United States.
doi:10.2172/1436065. https://www.osti.gov/servlets/purl/1436065.
```

```
@article{osti_1436065,
```

title = {A Minimum Variance Algorithm for Overdetermined TOA Equations with an Altitude Constraint.},

author = {Romero, Louis A and Mason, John J.},

abstractNote = {We present a direct (non-iterative) method for solving for the location of a radio frequency (RF) emitter, or an RF navigation receiver, using four or more time of arrival (TOA) measurements and an assumed altitude above an ellipsoidal earth. Both the emitter tracking problem and the navigation application are governed by the same equations, but with slightly different interpreta- tions of several variables. We treat the assumed altitude as a soft constraint, with a specified noise level, just as the TOA measurements are handled, with their respective noise levels. With 4 or more TOA measurements and the assumed altitude, the problem is overdetermined and is solved in the weighted least squares sense for the 4 unknowns, the 3-dimensional position and time. We call the new technique the TAQMV (TOA Altitude Quartic Minimum Variance) algorithm, and it achieves the minimum possible error variance for given levels of TOA and altitude estimate noise. The method algebraically produces four solutions, the least-squares solution, and potentially three other low residual solutions, if they exist. In the lightly overdermined cases where multiple local minima in the residual error surface are more likely to occur, this algebraic approach can produce all of the minima even when an iterative approach fails to converge. Algorithm performance in terms of solution error variance and divergence rate for bas eline (iterative) and proposed approach are given in tables.},

doi = {10.2172/1436065},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Sun Apr 01 00:00:00 EDT 2018},

month = {Sun Apr 01 00:00:00 EDT 2018}

}