Week 5 PART 2

Geology 2112 – Principles and Applications of Geophysical Methods WEEK 5

Lecture Notes – Week 5 PART 2

Reading: Fowler Chapter 5.4, 5.6.2

Objectives: Discuss the concept of the ‘Geoid’ and how we use it to describe the shape of the earth.

The Reference Ellipsoid (or Oblate Spheroid):

• So far we’ve seen the evolution of our understanding o f the shape of the earth from flat, to round (spherical) to the ‘oblate spheroid’ or ellipsoid • A ‘perfect’ ellipsoid can be characterized (relative to a sphere) by a ‘flattening parameter:

Where f is flattening, RE is the equatorial (longest) radius, and RP is the polar (shortest) radius

• Flattening of the earth spheroid is approximately 1/298

[ Note that Fowler introduces latitude corrections here, but we’ve already seen the concept with Richer’s clock, and we’ll return to the actual correction later]

BUT, the earth is not a perfect ellipsoid either – think of mountains, valleys, subduction zones, even small hills – these are all departures from the perfect ellipsoid

And we also still need a ‘reference surface’ or zero elevation to use as a starting point – traditionally we’ve used the mean or average sea‐level, which should be a perfect equipotential surface because it the water surface can flow and adjust to gravity

Equipotential Surfaces:

• We can simplify the shape of the earth down to perfect or reference ellipsoid • On the surface of that ellipsoid – or any imaginary concentric ellipsoid we choose – the force of gravity should be equal everywhere, and should be directed straight down (perpendicular to the surface) everywhere • That makes the surface of a perfect ellipsoid an ‘equipotential surface’ because gravity (gravitational potential) is the same everywhere

Geology 2112 – Principles and Applications of Geophysical Methods WEEK 5

• The term ‘equipotential’ refers to the potential energy associated with distance from the centre of mass – e.g. lift a book off of a table, you’ve done work against gravity and increased the gravtitational potential of the book • So an equipotential surface is one upon which gravitational potential is everywhere equal o Therefore, no work against gravity is done when moving along the surface o Gravity is directed perpendicular to the surface o The ‘vertical’ direction is parallel to gravity, perpendicular to the surface o The ‘horizontal’ direction in tangent to the curved surface • Note that for a sphere or ellipsoid, concentric smaller spheres or ellipsoids would all be equipotential surfaces, just with different values of the potential between them

On the oceans (far from land, no tides, winds, etc), the sea‐level is an equipotential surface. In geodetic surveying, we assume that the ellipsoid defined by sea‐level continues beneath the land, and we describe elevations of the land with reference to that assumed surface.

BUT, we still have a problem: it turns out that sea‐level is not parallel to a perfect ellipsoidal surface, although it is an equipotential surface (almost). The sea surface is also ‘lumpy’ just like the land surface, because the mass distribution within the earth is not uniform.

• The Geoid is that lumpy equipotential sea surface, extended through land. It’s an equipotential surface of the earth, and it’s not a perfect ellipsoid.

The GEOID:

The geoid is defined spatially by its departure height (N in figure above) from the reference ellipsoid. Can be positive or negative. A mass outside the ellipsoid, or an excess of mass below the ellipsoid leads to a positive geoid height. Less mass, lower geoid and negative height.

Geology 2112 – Principles and Applications of Geophysical Methods WEEK 5

Isostasy:

• 1737‐1740 Pierre Bouger noticed that his plumb‐bob vertical wasn’t deflected as much as expected by the mass of some big mountains in the Andes • 1806‐1843 George Everest compared triangulated distances with those determined astronomically (comparing plumb‐bob vertical with astronomical vertical, as Bouger did) and noticed the same effect as Bouger • Vertical (local gravity or the geoid surface) was only deflected about 1/3 of what was expected by all the extra mass above the ground • G.B. Airy in 1855 and J.H. Pratt in 1859 tried to explain the observations • The basically said that yes, there was a mass excess above the surface (above sea‐level or the reference ellipsoid), but there was also a mass defecit below somewhere • This led to the idea of deep mountain roots, and requires that the lithosphere is somehow ‘floating’ on a flowing mantle (asthenosphere) • The floating means that the lithosphere is less dense than the mantle, and that if we add a bunch of mass above the ‘surface’ (by thrusting or glaciations) the whole area must sink to compensate, and so less dense material is pushed into the mantle • Alternatively, if we remove a bunch of mass from a given location (e.g. melt the glaciers), the less dense material below floats up to a new equilibrium • Parts of the lithosphere that are out of equilibrium (being forced down or pushed up) will have gravity anomalies that don’t quite match the surface expression, and a positive or negative geoid anomaly results