NON-LINEAR LEAST SQUARES ESTIMATION AND ANALYSIS
TRILATERATION
1. Trilateration and Distance Observation Equation
o The distance observation equation is a fundamental equation used in trilateration,
which is a geomatics technique for determining the position of a point in space by
measuring its distances to a set of known reference points
o used in geomatics for various applications, such as surveying, mapping, and
navigation
o The distance observation equation relates the distances between a point and a set of
reference points to the coordinates of the point
o In two-dimensional trilateration, the distance observation equation can be expressed
as:
o d1 = sqrt((x - x1)^2 + (y - y1)^2) d2 = sqrt((x - x2)^2 + (y - y2)^2) d3 = sqrt((x - x3)^2 + (y
- y3)^2)
o where d1, d2, and d3 are the measured distances between the unknown point and the
reference points (x1, y1), (x2, y2), and (x3, y3), respectively. The unknown coordinates
of the point are x and y.
o The distance observation equation is nonlinear because the distances are functions of
the unknown coordinates.
o linearizing the equation using Taylor series expansion, a linear system of equations
can be obtained, which can be solved using least squares or other optimization
methods to obtain the estimated coordinates of the unknown point.
TRIANGULATION
1. Triangulation and Azimuth Observation Equation
o Triangulation:
Triangulation is a geomatics technique for measuring distances and angles
between reference points to determine the location of a point of interest.
It is based on the principles of trigonometry and geometry and is widely used
in surveying and mapping applications.
Triangulation involves measuring the angles between the reference points
using a theodolite or other precision instrument.
By combining the angle measurements with known distances between the
reference points, the location of the point of interest can be calculated using
trigonometric calculations.
o Azimuth Observation Equation:
The azimuth observation equation is used in triangulation to calculate the
horizontal angle (azimuth) between two reference points.
It is based on the law of cosines and relates the observed angle to the known
distances between the reference points.
The equation is given as: cos(A) = (b^2 + c^2 - a^2)/(2bc), where A is the
azimuth angle, and a, b, and c are the distances between the reference points.
The azimuth observation equation is particularly useful in geomatics
applications where precise horizontal positioning is required, such as in land
surveying or navigation.
It is often used in conjunction with other observations, such as vertical angles
or distances, to determine the location of a point of interest with high
accuracy.
TRILATERATION
1. Trilateration and Distance Observation Equation
o The distance observation equation is a fundamental equation used in trilateration,
which is a geomatics technique for determining the position of a point in space by
measuring its distances to a set of known reference points
o used in geomatics for various applications, such as surveying, mapping, and
navigation
o The distance observation equation relates the distances between a point and a set of
reference points to the coordinates of the point
o In two-dimensional trilateration, the distance observation equation can be expressed
as:
o d1 = sqrt((x - x1)^2 + (y - y1)^2) d2 = sqrt((x - x2)^2 + (y - y2)^2) d3 = sqrt((x - x3)^2 + (y
- y3)^2)
o where d1, d2, and d3 are the measured distances between the unknown point and the
reference points (x1, y1), (x2, y2), and (x3, y3), respectively. The unknown coordinates
of the point are x and y.
o The distance observation equation is nonlinear because the distances are functions of
the unknown coordinates.
o linearizing the equation using Taylor series expansion, a linear system of equations
can be obtained, which can be solved using least squares or other optimization
methods to obtain the estimated coordinates of the unknown point.
TRIANGULATION
1. Triangulation and Azimuth Observation Equation
o Triangulation:
Triangulation is a geomatics technique for measuring distances and angles
between reference points to determine the location of a point of interest.
It is based on the principles of trigonometry and geometry and is widely used
in surveying and mapping applications.
Triangulation involves measuring the angles between the reference points
using a theodolite or other precision instrument.
By combining the angle measurements with known distances between the
reference points, the location of the point of interest can be calculated using
trigonometric calculations.
o Azimuth Observation Equation:
The azimuth observation equation is used in triangulation to calculate the
horizontal angle (azimuth) between two reference points.
It is based on the law of cosines and relates the observed angle to the known
distances between the reference points.
The equation is given as: cos(A) = (b^2 + c^2 - a^2)/(2bc), where A is the
azimuth angle, and a, b, and c are the distances between the reference points.
The azimuth observation equation is particularly useful in geomatics
applications where precise horizontal positioning is required, such as in land
surveying or navigation.
It is often used in conjunction with other observations, such as vertical angles
or distances, to determine the location of a point of interest with high
accuracy.
