8.1 Ellipsoids, Geoid Undulation, Geodetic Datums, and the Philippine Reference System of 1992 (PRS92)

Ellipsoids, Geoid Undulation, Geodetic Datums, and PRS92

In geodetic surveying, positions must be referenced to well-defined mathematical surfaces. Because the Earth's physical surface is highly irregular and continuously undergoing dynamic change, three fundamental surfaces are defined in geodesy: the topographic surface, the geoid, and the reference ellipsoid.

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1. The Three Fundamental Geodetic Surfaces

Understanding the distinctions and mathematical relationships among these three surfaces is central to geodetic position and elevation determinations:

1. Topographic Surface: The actual physical surface of the Earth, including land masses, ocean beds, mountains, and valleys. It is highly irregular and cannot be expressed by a simple mathematical formula.
2. Geoid: The equipotential surface of the Earth's gravity field that best fits global mean sea level (MSL) in an undisturbed state (ignoring winds, tides, and ocean currents). The plumb line (direction of gravity) is everywhere perpendicular (normal) to the geoid. Because mass density distribution inside the Earth is non-uniform, the geoid contains irregular bumps and depressions relative to a mathematical surface.
3. Reference Ellipsoid: A mathematically defined oblate spheroid created by rotating an ellipse around its minor (polar) axis. It provides a smooth, regular surface upon which geometric computations of latitude ($\phi$), longitude ($\lambda$), distance, and direction can be performed.

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2. Geometry and Parameters of the Reference Ellipsoid

An oblate reference ellipsoid is defined mathematically by its semi-major axis $a$ (equatorial radius) and semi-minor axis $b$ (polar radius). Key derived geometric parameters include:

Primary Geometric Relationships

• Flattening ($f$): $f = \frac{a - b}{a}$
• First Eccentricity Squared ($e^2$): $e^2 = \frac{a^2 - b^2}{a^2} = 2f - f^2$
• Second Eccentricity Squared ($e'^2$): $e'^2 = \frac{a^2 - b^2}{b^2} = \frac{e^2}{1 - e^2}$

Principal Radii of Curvature

At any geodetic latitude $\phi$, the ellipsoid has two principal radii of curvature:

1. Radius of Curvature in the Meridian ($M$) (North-South direction): $M = \frac{a(1 - e^2)}{(1 - e^2 \sin^2 \phi)^{3/2}}$
2. Radius of Curvature in the Prime Vertical ($N$) (East-West direction, normal to the meridian): $N = \frac{a}{(1 - e^2 \sin^2 \phi)^{1/2}}$
3. Mean Radius of Curvature ($R_m$): $R_m = \sqrt{M N}$

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3. The Clarke 1866 Ellipsoid

The Clarke 1866 ellipsoid was derived by Alexander Ross Clarke and adopted historically for North America and the Philippines. Its defining dimensions are:

• Semi-major axis ($a$): $6,378,206.400\text{ m}$
• Semi-minor axis ($b$): $6,356,583.800\text{ m}$
• Flattening ($f$): $f = \frac{6,378,206.4 - 6,356,583.8}{6,378,206.4} = \frac{21,622.6}{6,378,206.4} \approx \frac{1}{294.9786982}$
• First Eccentricity Squared ($e^2$): $e^2 \approx 0.006768658$

In contrast, modern global geocentric systems like WGS84 use $a = 6,378,137.000\text{ m}$ and $f = 1 / 298.257223563$.

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4. Height Systems and Geoid Undulation

When positioning with Satellite Navigation Systems (GNSS / GPS), height is measured relative to the reference ellipsoid. Spirit leveling, however, measures height relative to the geoid (Mean Sea Level).

The Geoid-Ellipsoid Height Equation

$h = H + N \iff N = h - H$

Where:

• $h$ = Ellipsoidal Height (height above or below the reference ellipsoid measured along the ellipsoid normal).
• $H$ = Orthometric Height (physical elevation above Mean Sea Level / geoid, measured along the curved gravity plumb line).
• $N$ = Geoid Height (Geoid Undulation) (the vertical distance between the reference ellipsoid and the geoid).

          Topographic Surface (Ground)
                  /
                 /  | h (Ellipsoidal Height)
                /   | 
   ~~~~~~~~~~~~/~~~~|~~~~~~~~~~~~~~~~~~~~~~~~ Geoid (MSL, H = Elevation)
              /     | | N (Geoid Undulation)
-------------/------|-|---------------------- Reference Ellipsoid (h = 0)

[!NOTE] If the geoid lies above the reference ellipsoid at a given station, $N$ is positive ($N > 0$). If the geoid lies below the reference ellipsoid, $N$ is negative ($N < 0$). In the Philippines, relative to WGS84, geoid undulation $N$ generally ranges from $+40\text{ m}$ to $+55\text{ m}$.

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5. Luzon 1911 Datum vs. PRS92 Datum vs. WGS84 Datum

A geodetic datum defines the location, orientation, and scale of a reference coordinate system relative to the Earth.

Luzon 1911 Datum

Established during the American administration using classical triangulation:

• Datum Origin: Station Balanacan, located on top of a hill in Mogpog, Marinduque.
• Latitude of Origin ($\phi_0$): $13^\circ 33' 41.000''\text{ N}$
• Longitude of Origin ($\lambda_0$): $121^\circ 52' 03.000''\text{ E}$
• Azimuth to Station Baltazar: $9^\circ 12' 37.000''$
• Geoid Undulation at Origin ($N_0$): Assumed $N_0 = 0.00\text{ m}$ (geoid coincided with ellipsoid at Balanacan).
• Reference Ellipsoid: Clarke 1866.
• Classification: Non-geocentric local topocentric datum.

Philippine Reference System of 1992 (PRS92)

Due to regional distortion accumulation in classical triangulation and the advent of satellite positioning, the national network was re-observed using GPS from 1989 to 1991 under the Australian International Development Assistance Bureau (AIDAB) assistance.

• Establishment: Executive Order No. 45 (EO 45), signed on January 5, 1993 by President Fidel V. Ramos.
• Geometric Reference Ellipsoid: Retains the Clarke 1866 ellipsoid parameters for 2D horizontal position representation, but aligned with global satellite coordinates.
• Connection to WGS84: Aligned to WGS84 using a standard 7-parameter Helmert datum transformation: $\begin{bmatrix} X_{PRS92} \\ Y_{PRS92} \\ Z_{PRS92} \end{bmatrix} = \begin{bmatrix} \Delta X \\ \Delta Y \\ \Delta Z \end{bmatrix} + (1 + s) \begin{bmatrix} 1 & R_z & -R_y \\ -R_z & 1 & R_x \\ R_y & -R_x & 1 \end{bmatrix} \begin{bmatrix} X_{WGS84} \\ Y_{WGS84} \\ Z_{WGS84} \end{bmatrix}$

Executive Order No. 45 Mandates

1. Mandated PRS92 as the standard reference system for all land surveys, mapping, cadastral surveys, and geographic databases in the Philippines.
2. Designated the National Mapping and Resource Information Authority (NAMRIA) and DENR as primary implementing agencies.
3. EO 280 (August 14, 2000) and EO 321 (July 2, 2004) subsequently extended the transition timelines—EO 321 pushed the final integration deadline to the end of 2010—to allow complete migration of legacy land records into PRS92.

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6. Worked Examples

Worked Example 8.1.1: Calculating Ellipsoid Parameters

Problem: Given the Clarke 1866 semi-major axis $a = 6,378,206.400\text{ m}$ and semi-minor axis $b = 6,356,583.800\text{ m}$, compute the flattening $f$ and first eccentricity squared $e^2$.

Solution:

1. Compute flattening $f$: $f = \frac{a - b}{a} = \frac{6,378,206.400 - 6,356,583.800}{6,378,206.400} = \frac{21,622.600}{6,378,206.400} \approx 0.0033900753$ $1/f = \frac{1}{0.0033900753} \approx 294.9787$
2. Compute first eccentricity squared $e^2$: $e^2 = \frac{a^2 - b^2}{a^2} = \frac{(6,378,206.400)^2 - (6,356,583.800)^2}{(6,378,206.400)^2}$ $a^2 = 40,681,516,928,809.96$ $b^2 = 40,406,157,597,822.44$ $a^2 - b^2 = 275,359,330,987.52$ $e^2 = \frac{275,359,330,987.52}{40,681,516,928,809.96} \approx 0.006768658$

Worked Example 8.1.2: Orthometric Height Determination

Problem: A survey station observed with GNSS yields an ellipsoidal height $h = 152.480\text{ m}$ relative to WGS84. The geoid undulation model at this point gives $N = +46.230\text{ m}$. Calculate the orthometric height (elevation $H$ above MSL).

Solution:

1. Use the height relationship $h = H + N \implies H = h - N$.
2. Substitute the values: $H = 152.480\text{ m} - 46.230\text{ m} = 106.250\text{ m}$
3. The orthometric height (elevation above Mean Sea Level) is $106.250\text{ m}$.