1 Introduction

1.1 Data

In geophysics, we measure many different types of data depending on the physical method being used.

left: ERT/IP pseudosections for different frequencies
single values
often arranged logically
view on data important
sometimes associated to space

1.1.1 Data in seismology

time series of acceleration, velocity or position

1.1.2 Data in travel-time tomography

traveltimes between shot and geophones

1.1.3 Data in gravimetry

Bouguer anomaly of gravity profile in Zellwald

1.1.4 Data in electrical resistivity tomography (ERT)

current and voltage combinations

1.1.5 Data organization

can be a discretized function of space, time or frequency, and plotted as such (curves)
can depend on several sensor (shot/geophone, ABMN) and visualized as several curves or a coloured matrix (crossplot, pseudosection)

Data are subsumed in the data vector \[\textbf{d}=[d_1, d_2, \ldots, d_N]^T\] consist of true model response \(\bf f(m)\) plus noise \(\bf n\): \(\bf d=f(m)+n\)

1.2 Geophysical Workflow

Data acquisition
Preprocessing (QC, filtering)
Parameterization (mesh)
Inversion
Fit of data & model response
Postprocessing & visualization
Interpretation

1.3 Model

numerical parameterization of subsurface (our assumption)

1.3.1 Model types

subsurface described by model vector \(\textbf{m}=[m_1, m_2, \ldots, m_M]^T\)

Independent parameters:

seismology: earthquake location, stress, principal axis
gravity: position and depth of anomaly, size, density contrast
spectroscopy (e.g. SIP): function parameters (e.g. Cole-Cole)
layered model thicknesses and values (\(\rho\), \(v\))

subsurface described by model vector \(\textbf{m}=[m_1, m_2, \ldots, m_M]^T\)

Discrete functions of space, time, frequency

(same property aligned along axes)

refraction: depth (and velocity) of refractor
distribution of a parameter in space (and time)
spectroscopy: Fourier or Debye distribution
e.g. 2D/3D resistivity/velocity/density distribution

1.4 Inversion

Inverse problem: determine \(\vb m\) so that data are explained

\[ \vb d \approx \vb f(\vb m) \]

Why not

\[ \vb d = \vb f(\vb m) \]

Because \(\vb d\) contains random noise not to be explained.

1.4.1 Minimization problem

The (data) objective function

\[ \Phi_d=\|{\vb d - f(m)}\|^2_2 = \sum_i^N (d_i-f_i(\vb m))^2 \rightarrow \min \]

Take into account errors: Explain model within error (\(\boldsymbol{\epsilon}\)) bounds \[ \Phi_d = \| \frac{\vb d-f(m)}{\boldsymbol{\epsilon}}\|^2_2 = \sum_i^N \left(\frac{d_i-f_i(\vb m)}{\epsilon_i}\right)^2 \| \rightarrow \min \]

error model \(\boldsymbol{\epsilon}=[\epsilon_1, \epsilon_2, \ldots, \epsilon_N]^T\) (assumption of noise standard deviation)

1.4.2 Correctness

Well-posed problems (Hadamard)

There is a solution,
it is uniquely defined &
depends steadily from input data (small variations lead to small model deviations)

Ill-posed problem

There is no model to fit the data perfectly
Many models can fit the data within errors
Small data variations can lead to large model deviations

1.4.3 Occam’s razor - a fundamental rule

novacula Occami

Pluralitas non est ponenda sine neccesitate!

(William of Ockham, Scottish philosopher and theologian, 14th century)

Principle of Parsimony

Entities must not be multiplied beyond necessity.

Of two competing theories, the simpler explanation is to be preferred.

From all models fitting the data, choose the simplest!