Inverse Theory in Geophysics, Lecture 1

Inverse TheoryDefinition:Inverse Theory deals with the determination of the model parameters from the data. In our case it is the estimation of subsurface structure from surface measurements.

data Model Model parameters

DefinitionA model may be defined as the mathematical relation (either analytical or empirical) that relate the independent variable X with the dependent variable Y in the form:

Y= a + b XIn this model, the model parameters are a and b. In algebraic notation they are the unknowns of the equation. In matrix form this relation may be written as:

A m = d

Where:

A Is the kernel or coefficients matrix

m is the vector of model parameters a & b

D is the data vector

(1 𝑥1⋮ ⋮1 𝑥𝑛

)(𝑎𝑏)=(𝑌 1

⋮𝑌 𝑛

)

The model here for gravity field is:

To define the theory in more practical way we use the beach example where we expect a pirates treasure is burried under the sand. The treasure is made of gold which is much more dense than sands. Thus we propose to use the gravity method.

The problem is that we

Thus the methodology here is forward modeling not inverse. The ideal model of the beach may be sketched as:

The computation algorithm we will adopt will

As we have 45 blocks we end up with:

We also end up with no unique solution.

The models may be categorized as:

Problems we face:

Why!!!Because the beach

End of Lecture

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