# Isostasy

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Sometimes the materials linked to below will also be handed out in class in hard copy. But, for anything you need to read it will be linked to here for each "topic" we cover.

Gravity from the book Looking Into The Earth (CH 8)

Isostasy from the book Looking Into The Earth (CH 9)

Why are the High Plains High?

## Isostasy

Thursday 2021-09-23

### Demo of measuring gravitational acceleration.

Data collected in class - will be linked to here.

### Gravity with and without a mountain "root"

If mountains were situated as shown below, we would expect find a gravity increase (larger with a bigger mountain) compared to the situation without a mountain:

Now think of a mountain with a low-density "root". Low-density compared to the crust surrounding it:

An actual measurement of gravity would show a much smaller increase than you would expect. This could be explained via a low-density mountain root.

### Archimede's Principle

This leads us to think about the crust and lithosphere as "floating" on a higher density feature (the asthenosphere) that at some depth compensates for all the weight. This is where Archimede's Principle comes into play when making a model for this phenomena. Note how the weight of the object when in equilibrium is supported by the water (or, in the earth's case, asthenosphere):

Reminder of the terms crust, lithosphere, asthenosphere:

If compensated, gravity should be approximately constant:

## The equations of isostatic calculations

When "columns" (e.g. idealized mountain ranges, sedimentary basins) are in isostatic equilibrium then the pressure they exert at their compensation depth is the same (the handout I gave you refers to this aw equal weight):

Similarly you can equate heights of the columns:

Compare these equations in the context of the figure below:

Warning: Be careful with the subscripts - sometimes the subscript "a" is used for "air", sometimes it is used for "asthenosphere, or asth". So be careful you you what ${\rho }_{a}$$\rho_a$ or ${h}_{a}$$h_a$ is actually referring to. To be safe, you should probably in your own work write out things explicitly as ${\rho }_{air}$$\rho_{air}$, ${h}_{air}$$h_{air}$ and ${\rho }_{asth}$$\rho_{asth}$, ${h}_{asth}$$h_{asth}$.

### Example: Adding ice to a land surface

After adding 2 km of ice to a continent (which wasn't all that long ago!!), what is the height of the top of the ice compared to the original surface?

### Example: Filling in a lake with sediment

Papers that discuss the history of the surveys finding the gravity anomalies near mountains, as well as the development of Airy and Pratt isostasy models.

Pratt, 1854

Airy, 1855

Dutton, 1889

## Measuring Gravity (and Corrections)

We have seen that gravity anomalies exist (i.e. gravity varies from place to place). How large are these variations relative to the "typical" value of $9.8\frac{m}{{s}^{2}}$$9.8 \frac{m}{s^2}$? How will these anomalies be represented quantitatively?

Imagine a buried "density" anomaly as shown below:

At location C a measured gravity value should be larger than at location P because the density anomaly (increase in this case) is closer to the C whereas at P the the anomaly is further away.

An anomaly has the following effect on a gravity reading:

so to determine the gravity anomaly take as many closely spaced readings as you can (imagine the anomaly is a large ore body you are interested in prospecting - see Table 8.1 for how much denser ore bodies are than the surrounding shallow crust!):

See the nature of the gravity anomaly due to shallower vs deeper half sheets:

Half-sheets generated due to faulting.

### Topographic corrections to gravity readings

We don't measure on flat surfaces!!!

#### Free-air correction

Consider $A↑B$$A \uparrow B$. g decreases by $0.3086\frac{mGal}{m}$$0.3086 \frac{mGal}{m}$ rise. So the elevation changes along the surface are important.

#### Bouguer correction

Now consider $B\to C$$B \rightarrow C$. The extra mass of the plateau needs to be accounted for (via a slab model)

$\delta g_{Bouguer} =2*pi*G*\rho *h \space\space mGal$.

Bouguer is SUBTRACTED.

Free-air and Bouguer can be combined: $\delta g = h*(0.3086-0.04192*\rho) \space\space mGal$

#### Terrain Correction

Consider point $D$$D$. Here the local variations in the surface can't be accounted for by a slab model.

H creates a lateral and upward pull, reducing g. V removes a downward pull, so also reduces g. Therefore these are added to the measurement of g.

#### The Bouguer Anomaly

Bouguer Anomaly = measured/observed value of g + free-air correction - Bouguer correction + Terrain correction - latitude correction + Eotovos correction.

#### The point

Gravity survey values need modifications to isolate the subsurface anomaly from the other anomalies at the surface. This allows one to properly interpret the subsurface anomaly for geologic questions and/or resource exploration.

## The Isostatic Anomaly

How can we tell if a region is in isostatic equilibrium?

## Other Examples

Going out and testing some topographic features:

• Are the Rocky Mountains Isostatically Supported?
• Why are the high plains high?