Coordinate Reference Systems for Geodetic Surveying with Geomatics and R

Marcelo de Carvalho Alves and Luciana Sanches
"2021-05-10"

Introduction

The datum was a term that meant geometric reference in geomatics and has been used to define a referential for position of geometric elements in space or in a topographic mapping.
The set of information defining shapes, size, origin and orientation of coordinate system established for the positioning of points on the Earth's surface geodetic with reference by geodetic datum.
After determining and deploying a geodetic datum for a region or a country, a network of points with coordinates referenced to that datum must be deployed, called the geodetic reference system.

Introduction

Geodetic surveys have been used to establish horizontal and vertical accuracy of reference monuments and have been used as a basis for generating or checking survey projects such as topographic and hydrographic mapping, boundary demarcation, route definition, construction planning, among others.
The established references have been essential as support for the use of geographic and Earth information systems.

Introduction

Types of geodetic datum:

Horizontal: Planimetric positions of points on the Earth's surface established through latitude, geodetic longitude, direction and parameters of the reference ellipsoid.
Vertical:References established to determine orthometric altitudes of points over large areas.

Introduction

Different methods can be used to define a network of geodetic control points in order to extend the understanding about the shape and mass of the Earth.
Extending the network of geodetic control points can be accomplished by leveling, accurate polygonals, very long base interferometry, satellite laser mapping, GNSS observations, and gravity surveys.

Introduction

The accumulation of these data can be used to update and extend existing control data, as well as to accurately define the geometric shape of the Earth.
GNSS surveying can also be used to establish vertical control, but with limitations due to the need to obtain an accurate geoidal model as a reference.

Introduction

Control grid reference for points

Topographic mapping and large-scale production planning
Dimension control in construction
Surveying deformation of structures
Augmentation and densification of existing control networks

Introduction

Geodetic reference system establishment

Topographic surface: Irregularities of the relief, mountain ranges, valleys, fields, oceanic hollows and marshes.
Real Earth's shape: The geoidal surface, determined as a function of the equipotential surface of the Earth's gravitational field, considered as the .
Ellipsoid of revolution: The geometric surface, determined by a sphere flattened at the poles.

Geoid and Ellipsoid

The ellipsoid was defined as a mathematical surface obtained by the revolution of an ellipse on the Earth's polar axis. The dimensions of the ellipsoid were selected to determine a good fit of the ellipsoid on the geoid over large areas, based on surveys conducted in the area. The ellipsoid approximated the geoid and can be mathematically defined to calculate positions of points separated by large distances in control and geodetic surveys [@Alves2016].

Geoid and Ellipsoid

Illustration of setting ellipsoid and geoid on the Earth (left) and an equipotential surface (right).

Illustration of setting ellipsoid and geoid on the Earth (left) and an equipotential surface (right).

Geoid and Ellipsoid

The first accurate terrestrial ellipsoid was determined by German astronomer Friedrich Wilhelm Bessel (1784-1846) in 1841.

Ellipsoids considered in Brazil

International Ellipsoid of 1924 (Hayford), Córrego Alegre datum.

Reference Ellipsoid of 1967, Chuá datum, South American Datum of 1969 (SAD-69).
Ellipsoide Geodetic Reference System of 1980 (GRS-80), Geocentric Reference System for the Americas (SIRGAS-2000).

Geoid and Ellipsoid

Table: Definition of ellipsoid parameters used in geomatics.

Ellipsoid	Semi-axis a (m)	Semi-axis b (m)	Flattening f	Use
GRS-80	6378137.0	6356752.314140356	1/298.25	Global (ITRS), Brasil, USA
WGS-84	6378137.0	6356752.314245179	1/298.25	Global (GPS)
Hayford-1924	6378388.0	6356911.946127947	1/297.00	USA / Europe
Córrego Alegre	6378388.0	6356911.946127947	1/297.00	Brasil
GRS-1967	6378160.0	6356774.516000000	1/298.24	Australia
SAD-1969	6378160.0	6356774.719000000	1/298.25	South America

Geoid and Ellipsoid

The flattening of the ellipsoid can be described by:

\begin{align} f=1-\frac{b}{a}=\frac{a-b}{a} \end{align}

where \( a \), was half of the major axis and, \( b \), half of the minor axis of the Earth's ellipsoid of revolution around the \( PP' \) axis.

Parameters a and b of an ellipse used to characterize the Earth's shape.

Parameters a and b of an ellipse used to characterize the Earth's shape.

Geoid and Ellipsoid

Ellipsoid first eccentricity (\( e \)) and first quadratic eccentricity (\( e^2 \))

\[ e=\frac{\sqrt {a^2-b^2}}{a}=\frac{a^2-b^2}{a^2}= \sqrt{2f-f^2}=\sqrt{f(2-f)} \]

\[ e^2=\frac{a^2-b^2}{a^2}=2f-f^2 \]

Geoid and Ellipsoid

Second eccentricity (\( e' \)) and second quadratic eccentricity (\( e'^{2} \))

\[ e'=\frac{\sqrt {a^2-b^2}}{b}=\frac{e}{\sqrt{1-e^2}}= \frac{e^2}{1-e^2}=\frac{f(2-f)}{(1-f)^2} \]

\[ e'^2=\frac{a^2-b^2}{b^2}=\frac{e^2}{1-e^2} \]

Geoid and Ellipsoid

By assuming the ellipsoid as a sphere, the radius and volume of the reference ellipsoid were equated to those of the sphere:

\[ r=\sqrt[3]{a^2b} \]

Geoid and Ellipsoid

The spherical distance between two points on the Earth's surface that are at the same distance from the center of the spheroid representing the Earth was measured over the arc generated by the spheroid containing the two points.
In practice, reductions from topographic to ellipsoidal distances were performed by considering the ellipsoid as a spheroid of radius equal to the local mean radius of the Earth at the latitude of the point.
It may be irrelevant whether the Earth is a spheroid or an ellipsoid for conventional topographical calculations .

The Conventional Terrestrial Pole

The ellipsoid was defined on the basis of the size of an ellipse rotated around the Earth's polar axis.
Polar axis motion can be subdivided into two categories called precession and nutation.
By international convention, the Earth's mean axis of rotation was defined between the years 1900 and 1905 and called the Conventional Earth Pole (CTP).

The Conventional Terrestrial Pole

Conventional Earth system with nutation and precession movements of the Earth's polar axis.

Conventional Earth system with nutation and precession movements of the Earth's polar axis.

Geodetic Position and Ellipsoidal Raddi

Definition of different radii and a vertice P on the ellipsoid.

Definition of different radii and a vertice P on the ellipsoid.

Geodetic Position and Ellipsoidal Raddi of Curvature

Meridians were large circles on the circumference of the ellipsoid that passed through the north and south poles.
The plane defined by the vertical circle passing through the vertice \( P \), perpendicular to the meridian plane in the ellipsoid, was called the main vertical or normal section.
The radius of the main vertical at point \( P \), \( R_N \), was called the normal, because it was perpendicular to a plane tangent to the ellipsoid at \( P \).

Geodetic Position and Ellipsoidal Raddi of Curvature

The geodetic latitude \( \phi_P \) was the angle in the meridian plane containing \( P \) between the equatorial plane and the normal at \( P \).
The geodetic altitude \( h_P \) should be included to define the location of point \( P \) on the Earth's surface.

Geodetic Position and Ellipsoidal Raddi of Curvature

The geodetic altitude was the distance measured along the length of the normal from \( P' \) on the ellipsoid to \( P \) on the Earth's surface.
The geodetic altitude was not equal to the elevation determined by differential leveling.
The great circle that defined the prime vertical at \( P \) had a radius of the normal section \( R_N \) that differed from the radius at the meridian \( R_M \) at \( P \).

Geodetic Position and Ellipsoidal Raddi of Curvature

The radius \( R_\alpha \) of the great circle at any azimuth \( \alpha \) in relation to the meridian was:

\[ R_N=\frac{a}{\sqrt{1-e^2sen^2\phi}}=R_M(1+e'^2cos^2\phi) \]

\[ R_M=\frac{a(1-e^2)}{(1-e^2sen^2\phi)^{3/2}} \]

Geodetic Position and Ellipsoidal Raddi of Curvature

\[ R_\alpha=\frac{R_N R_M}{R_Ncos^2\alpha+R_Msen^2\alpha} \]

where, \( a \) and \( e \) were parameters of the ellipsoid, \( \phi \) was the geodetic latitude of the station at which the radius was calculated.

Geodetic Position and Ellipsoidal Raddi of Curvature

The local mean radius of the Earth's curvature can be understood as the radius of a sphere that tangent the reference ellipsoid at the point of interest. The mean radius was calculated by geometric averaging the radius at the meridian and the radius of the normal section:

\begin{align} R_0=\sqrt{R_MR_N} \end{align}

Geoid Undulation and Deflection of the Vertical

Measurements with topographic equipment can be made relative to the ellipsoid or the geoid, yielding different results.
Therefore, it was important to know the conceptual differences between height, altitude, normal line and vertical line of a place.

Geoid Undulation and Vertical

Height: Dimension of a body from the base to the top
Altitude: Value of the elevation of a point relative to a vertical datum

Normal line: Line that intercepts the topographic surface, perpendicular to the ellipsoidal surface
Vertical line: Tangent line to the line of gravitational force that intercepted the topographic surface, perpendicular to the geoidal surface and that can be materialized by a plumb line.

Geoid Undulation and Vertical

The separation between geoid and ellipsoid determined the difference between the height of the point above the ellipsoid (geoidal altitude) and above the geoid (orthometric altitude), known as elevation.
This difference, called geoidal altitude or geoidal separation, can be observed when comparing the geodetic altitude of a point obtained by GNSS survey with the elevation determined by leveling.

Geoid Undulation and Vertical

Relationship between the orthometric altitude (\( H \)) and the geodetic altitude (\( h \)) at a point

\begin{align} h=N+H \end{align}

where, \( N \) was the geoidal altitude.

Relationship between the ellipsoid and the geoid.

Relationship between the ellipsoid and the geoid.

Geoid Undulation and Vertical

The projected components of the total vertical deviation in the meridian and normal planes were denoted \( \xi \) and \( \eta \), along the NS and EW directions, respectively.

Components of the vertical deflection.

Components of the vertical deflection.

Geoid Undulation and Vertical

The components, \( \xi \), on the meridian line and, \( \eta \), perpendicular to the meridian section, can be calculated by:

\begin{align} \xi=\phi_A-\phi_G \end{align}

\begin{align} \eta=(\lambda_A-\lambda_G)cos\phi=(Az_A-Az_G)cot\phi \end{align}

Geoid Undulation and Vertical

Laplace's equation can be derived from this equation by:

\begin{align} Az_G=Az_A-(\lambda_A-\lambda_G)sen\phi=Az_A-\eta tan\phi \end{align}

where, \( \phi_A \) and \( \phi_G \), were the astronomical and geodetic latitudes, \( \lambda_A \) and \( \lambda_G \), the astronomical and geodetic longitudes, \( Az_A \) and \( Az_G \), the astronomical and geodetic azimuths, respectively.