an introduction to inverse problems

8/10/2019 An introduction to inverse problems

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Understanding inverse theory Ann. Rev. Earth Planet. Sci ., 5, 35-64, Parker (1977).

Interpretation of inaccurate, insufficient and inconsistent data

Geophys. J. Roy. astr. Soc., 28, 97-109, Jackson (1972).

Monte Carlo sampling of solutions to inverse problems

J. Geophys. Res., 100, 12,431–12,447,

Mosegaard and Tarantola, (1995)

Monte Carlo methods in geophysical inverse problems,

Rev. of Geophys., 40, 3.1-3.29,Sambridge and Mosegaard (2002)

Some papers:

There are also several manuscripts on inverse problems availableon the Internet. I can not vouch for any of them.

See http://www.ees.nmt.edu/Geop/Classes/GEOP529_book.html



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Inferring seismic properties of the Earth’s interior

from surface observations



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When data only indirectly constrain quantities of interest



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What is that ?

What can we tell about

Who/whatever made it ?

Measure size, depthproperties of the ground

Collect data:

Who lives around here ?

Use our prior knowledge:

Make guesses ?

Can we expect to reconstruct thewhatever made it from the evidence ?



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Courtesy Heiner Igel

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Gravimeter

Hunting for gold at the beach with a gravimeter



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Gravimeter

We have observed some values:

10, 23, 35, 45, 56 µ gals

How can we relate the observed gravity

values to the subsurface properties?

We know how to do the forward problem:

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)'()( dV

r r

r Gr ! "=# $

This equation relates the (observed) gravitational potential to thesubsurface density.

-> given a density model we can predict the gravity field at the surface!



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What else do we know?

Density sand: 2.2 g/cm3

Density gold: 19.3 g/cm3

Do we know these values exactly ?

Where is the box with gold?

X

One approach is trial and error forward modelling

Use the forward solution to calculate many models for a rectangular boxsituated somewhere in the ground and compare the theoretical (synthetic )data to the observations.



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But ...

... we have to define plausible modelsfor the beach. We have to somehowdescribe the model geometrically.

We introduce simplifying approximations

- divide the subsurface into rectangles with variable density- Let us assume a flat surface

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x x x x xsurface

sand

gold



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X

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This is called a priori (or prior) information.It will allow us to define plausible, possible, and unlikely models:

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plausible possible unlikely

Is there anything we know about the

treasure?

How large is the box?

Is it still intact?

Has it possibly disintegrated?

What was the shape of the box?



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X

XX

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Gravimeter

X

Things to consider in formulating the inverse problem

Do we have errors in the data ?

Did the instruments work correctly ?

Do we have to correct for anything?

(e.g. topography, tides, ...)

Are we using the right theory ?

Is a 2-D approximation adequate ?

Are there other materials present other than gold and sand ?

Are there adjacent masses which could influence observations ?

Answering these questions often requires introducing more

simplifying assumptions and guesses.

All inferences are dependent on these assumptions. (GIGO)



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Models with less than 2% error.



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Models with less than 1% error.



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Inverse problems = inference about physical

systems from data X

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-

Data usually contain errors (data uncertainties)-

Physical theories require approximations-

Infinitely many models will fit the data (non-uniqueness)-

Our physical theory may be inaccurate (theoretical uncertainties)-

Our forward problem may be highly nonlinear

- We always have a finite amount of data

Detailed questions are:

How accurate are our data?

How well can we solve the forward problem?

What independent information do we have on the model space

(a priori information) ?



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Often continuous functions are discretized to produce a finite set of

unknowns. This requires use of Basis functions

become the unknowns

are the chosen basis functions

All inferences we can make about the continuous function

will be influenced by the choice of basis functions. They

must suit the physics of the forward problem. They bound

the resolution of any model one gets out.



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Example of Basis functions

Local support

Global support



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Terminology can be a problem. Applied mathematicians oftencall the equation above a mathematical model and m as itsparameters, while other scientists call G the forward operator

and m the model.

Mark 2



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What is the forward problem ?

What is the inverse problem ?

an introduction to inverse problems

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