# 3.5: Satellite-derived Gravity and Geoid

- Page ID
- 3523

Recall the equation:

\[f_{hyd}=\frac{\frac{5}{2}\frac{\omega^2a^3}{GM}}{1+(\frac{25}{4})(1-\frac{3}{2}\frac{C}{Ma^2})^{\frac{1}{2}}}\]

In section 3.4, we wanted to solve for \(\frac{C}{Ma^2}\), substitute values for f_{hyd }(J_{2}), \(\omega\), a, and M, and calculate \(\frac{C}{Ma^2}\) for the planet.

But how do we actually find J_{2} and therefore U?

##### Finding U

We can use orbits to measure J_{2}, and we know that each reference surface of the ellipsoid has a constant value for U. We need to choose one close to the actual surface of the planet⇒this is the reference ellipsoid.

For earth, we can choose sea level. **FIGURE sea level**

- There is a layer of water→ocean
- U is parallel to surface, g is perpendicular
- Add a mass anomaly
- U moves out
- g is deflected→horizontally
- Water flows in
- Sea surface follows U
_{obs} **FIGURE U**_{obs}- Land and water
- U
_{ref}→cuts through these - U
_{obs}→undulations due to mass - U
_{obs}-U_{ref}→geoid height→N

+N over +m and -N over -m. |N|→\(\pm\)100

**Beware**...

Slope usage ie geoid surface\(\neq\)equipotential surface, but it is often used to mean this. Also, geoid height=geoid anomaly.

Before we said use orbits to get U_{obs}→U_{ref-ellipse}=geoid height

\[U\propto\frac{Gm}{r}+\frac{1}{r^3}+...\]

Smaller scale features on the surface are harder to measure. But what if you could measure U at sea level? This is called satellite altimetry. We can compare satellite gravity (altimetry) to geoid height. This can show us lots of smaller scale features such as seamounts.

**FIGURE Atlantic ocean ****topography **If we look at the below picture of the Atlantic, what is the biggest topographic signal?

⇒The continents vs the oceans. There is no step function in the zone at the edge, and instead we see a "ring". Why?

**FIGURE U brick +m and -m**

*Smaller features *

- seamounts
- fracture zones
- trenches
- These are all compensated and supported by the strength of the lithosphere
- ⇒gravity anomaly

Bigger features

- continents
- mountain ranges
- compensated or "floating"
- supported by viscous stresses, often mantle has flowed away⇒isotasy
- ⇒no gravity anomaly

Hudson Bay

Let's look at an example of Hudson Bay during the last ice age. During the ice age, the bay had a kilometers thick ice cap sitting on top. This is a compensated example because the lithosphere deforms. Flashing forward to the bay today, the ice has melted and the lithosphere has slowly rebounded from its deformed state. It is now in an uncompensated state. An important thing to note is that the bay may never fully return to its undeformed state, as the bay is filling in with sediments.

**FIGURE compensated**

Subduction Zones

**FIGURE +m subduction zones**

The mountain range is compensated⇒no \(\Delta g\) or N. The slab is not compensated. However, determining this can be a bit tricky.

**FIGURE tricky**

In earth⇒viscous layers

**FIGURE Drag**

The lithosphere is highly viscous ie cold. The upper mantle/asthenosphere is weak and has a low viscosity. The lower mantle, due to different minerals, has a high viscosity.