EART20170 Computing, Data Analysis & Communication skills

Lecturer: Dr Paul Connolly (F18 – Sackville Building)p.connolly@manchester.ac.uk

2. Computing (Excel statistics/modelling)2 lecturesassessed practical work

Course notes etc: http://cloudbase.phy.umist.ac.uk/people/connolly

LAST LECTURE!

Recommended reading: Cheeney. (1983) Statistical methods in Geology. George, Allen & Unwin

Plan

This lecture plus two more drop-in sessions in computer labs

Assessment handed out today and need to hand in by 16:00, Tuesday December 12th.

Lecture 5

Monte Carlo method of error propagation.

Using `Goal seek’ to root-find

Using `solver’ for optimisation

Basic macros.

Mega Tsunami

Statistical approach to error propagation

Computers enable the use of a very simple statistical method to propagate errors.

Monte Carlo methods provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments.

The statistical approach is particularly useful for propagating errors in complex functions.

Monte Carlo methods

Monte Carlo simulations or methods are named after Monte Carlo, Monaco, where the primary attractions are casinos containing games of chance exhibiting random behaviour.

The random behaviour in games of chance is similar to how Monte Carlo simulation selects variable values at random to simulate a model.

For each uncertain variable (one that has a range of possible values), you define the possible values with a probability distribution (e.g. the Excel function norminv(rand(),mean,stdev)).

Monte Carlo in Error Propagation

Let’s use a previous example of measuring bed thickness. We have two “populations” of measurements: x = 12.1 ± 0.3 and y = 4.2 ± 0.2.

By repeatedly taking samples at random (e.g. by the `nested’ Excel function norminv(rand(),mean,stdev)) from x and y, and adding the values, we should obtain a third population with a mean of 16.3cm and a standard deviation approximately equal to that obtain from the analytical solution (± 0.4 cm).

The statistical approach is particularly useful for propagating errors in complex functions

2t

222RJ JR

11

T

tJ

RJ

1t)exp(

This is one used in geochronology (don’t worry about the details):

The error propagation formula is given by:

where T = 1/

A more complicated formula:

Using the following data:

R = 49.704 ± 0.381t = 1.072 ± 0.011 billion years = 5.543 10-10 years-1

Then using the equations:

J = 0.016329 ± 0.000255

OK so what about the Monte Carlo?

A more complicated formula:

With a table of R and t calculated from the norminv function (10000 values are typically used for good statistics) we calculate J and can therefore calculate the average and stdev.

Using `Goal seek’ to root-find

What if I want to find the inverse of a function?

0 1 2 3 4 5 6

-5

0

5

10

15

20

25

x

f(x)

A typical function

?

Sometimes I can find the inverse analytically, e.g.

10

102

yx

xy

But not always (and if maths isn’t your forte).

Using `Goal seek’ to root-find

What height of fall will result in a height of single tsunami 30 m?

rL

HD

LHDg

E wt

2directionnpropagatiotolarperpendicuwaveofLength

shorenearheightwave33

8 3

rL

HD

Lg

EHD

w

t

2directionnpropagatiotolarperpendicuwaveofLength

shorenearheightwave

8

333

This is sometimes difficult (or impossible!).

Instead use iteration => Goal seek.

Using `Goal seek’ to root-find

Here are some arbitrary values

Go to tools->goal seek

Using `Goal seek’ to root-find

Enter the cell you want to change and the value (i.e. the actual energy) and the variable that will be changed – press OK

The cells change until the goal is found. Press OK at the next prompt

Your value for HD is now displayed in the correct cell. And you didn’t have to do any maths!

Using `solver’ for optimisation

Goal seek only works for functions of one variable.

Goal seek is good for route finding, but what if I want to find other properties such as minimum, maximum values?

E.g. Mining a gold seam. How can I break even? Whats the max profit I can make? Whats the min number of days I can mine before making a profit?

Using `solver’ for optimisation

Example: say that it costs £100 per day to hire your basic digging equipment.

And you manage to extract 4 tonne per day of gold from rock.

But as the number of days increase it becomes more difficult to extract the gold from the shaft as extra equipment has to be rented – usually have some a-priori knowledge (0.2xday^2).

The market value for gold is £321 for 31.1g.

Using `solver’ for optimisation

You wish to know: How many days you should work

before breaking even? What is the maximum net profit

you can make How long can you work before

your net rate of pay drops below £40 per day

Using `solver’ for optimisation

First: How many days to break even?

Go to Tools->solver

On the pop-up menu, set the target cell to the `Net’ cell reference and the changing cell to the `Days’ cell reference. Also check the `Value of’ tab and set this value to 0 (i.e. break even)

First: How many days to break even?

You should also set the constraint that the number of Days is greater than or equal to zero! – Click on `add’ and in the next box put in that `Days’ should be greater than 0 – OK.

On the first popup window press `solve’

The cell values change and another popup asks if you want to keep the solution – OK.

First: How many days to break even?

You see that it after 243 days the venture will start to become non profitable. Your total costs were £36139 all of which you got back from the gold seam.

Second: What is the max net profit?

Go to Tools->solver

On the pop-up menu, set the target cell to the `Net’ cell reference and the changing cell to the `Days’ cell reference. Also check the `max’ tab.

Second: What is the max net profit?

You should also set the constraint that the number of Days is greater than or equal to zero! – Click on `add’ and in the next box put in that `Days’ should be greater than 0 – OK.

On the first popup window press `solve’

The cell values change and another popup asks if you want to keep the solution – OK.

Second: What is the max net profit?

You see that it takes 121.6 days to get the maximum net profit of £2960. Your total costs were £15113 and your average rate of pay was £24 per day.

Third: At least £40 per day?

Go to Tools->solver

On the pop-up menu, set the target cell to the `Rate’ cell reference and the changing cell to the `Days’ cell reference. Also check the `Value of’ tab and set this value to 40 (i.e. £40/day)

Third: At least £40 per day?

You should also set the constraint that the number of Days is greater than or equal to zero! – Click on `add’ and in the next box put in that `Days’ should be greater than 0 – OK.

On the first popup window press `solve’

The cell values change and another popup asks if you want to keep the solution – OK.

Third: At least £40 per day?

You see that after 43 days your average net rate of pay will drop below £40. Your total costs were £4687 and your net pay was £1730.

Basic macros

The goal seek and solver tools are very powerful, but they can be time consuming if you want to work on vast data sets.

You can save a macro to a worksheet and use it again and again without having to always remember the exact sequence.

We will look at recording and using a macro for using the goal seek tool.

Basic macros

Go to tools->Macro->record new macro

You can name the macro and give it a shortcut key

Basic macros

The macro recorder is now visible with a `stop’ symbol. All your actions will now be recorded.

Again use goal seek in the same way as before

Basic macros

Enter your values as before.

The solution is found

Now press the `stop’ button to cease recording

Basic macros

You can now run your macro by going to tools->macros->macros

Selecting the macro you recorded and pressing run. You could have also used a shortcut

A subtlety

When using goal seek it is nearly always more convenient to solve for a zero.

This is because `goal seek’ doesn’t allow the value to be input by a cell reference.

A subtlety

In this case you put zero in the `To value’ box in goal seek

Put zero here

Mega tsunami

Volcano collapse

All volcanoes are inherently unstable and edifice growth will ultimately lead to some degree of collapse.

Major collapse of the old volcanic edifice, Soufriere Hills volcano early on 26 December 1977

Caldera collapse

The movement associated with collapse can be either vertical (caldera) or horizontal (lateral collapse).

The landslide moved northward at speeds of 110 to 155 mph and advanced . Part of the avalanche surged into and across Spirit Lake, but most of it flowed westward along the North Fork of the Toutle River for 13 miles filling the valley to an average

depth of 150 ft.

http://pubs.usgs.gov/publications/msh/debris.html

Mount St Helens 1980

Hazard Potential

Lateral collapse of oceanic island volcanoes are amongst the most spectacular natural events on Earth.

There is a potential for submarine landslides to generate tsunami and mega-tsunami.

Mega-tsunami have never been witnessed historically and geological evidence for their existence is controversial.

With ~1% of the world’s population (~60,000,000 people) living in regions susceptible to giant waves around the coastlines of the world’s oceans, they pose a very serious threat.

Mega-tsunami

Mega-tsunami are long wavelength (typically 300-400 km) wave trains that travel thousands of kilometres, across ocean basins at velocities in excess of 500 km hr-1.

As they pass into shallower water towards land their wavelength is compressed and height amplifies, typically 10- to 20-fold, generating waves up to hundreds of metres high that may incur many kilometres inland.

USGS http://vulcan.wr.usgs.gov/Volcanoes/Hawaii/Maps/map_location_hawaii.html

Hawaiian lateral collapses The Hawaiian islands are surrounded by more

than 68 slumps and avalanches >20 km long. There are >20 giant collapses of up 5000

km3 (approx. 2000 times larger than Mt St Helens)

From: http:/www.mala.bc.ca/~earles/kilauea-feb02.htm

From: Ward, 2002

Lateral collapse

10 kmLanai

Molokai

N

USGS http://vulcan.wr.usgs.gov/Volcanoes/Hawaii/Maps/map_location_hawaii.html

Prehistoric Hawaiian Collapse

PACIFIC OCEAN

Wave impact

HAWAII

NEW GUINEA

NEW ZEALAND

AUSTRALIA

FIJI

Sydney

Source of tsunami

TASMANIA

Wave impact

Wave impact

From Davidson, 1992

Lanai tsunami impact

New South Wales Tsunami Deposits

Blocks stacked against against 30 m high cliffs. Note the person circled for scale. Some of the blocks are as large as rooms in a house.

The tsunami carved these scour pools within a few minutes as it overtopped a 20-25 m high headland

Source: E.A. Bryant http://www.uow.edu.au/science/geosciences/research/tsun.htm

Potential energy released by the collapse:

)()( 10 DDgVE wrp

V = volume of collapse block (m3)pw = density of seawater (1030 kg m3)pr = density of rock (2800 kg m3)g = acceleration due to gravity (9.8 m/s/s)D0 = initial depth of sliding block (m)Ds = final depth of sliding block (m)

D0

Ds

Tsunami wave model

Archimedes force

LHD33

g8E 3w

t

HD ~ H, the wave height near shore (Depth ~ 0)

L = length of wave perpendicular to the propagation direction

wave L = 2r

r

The wave energy, Et