BRIEF DISCUSSION ON INVERSE THEORY

WEAKLY SEMINAR, PPSF@USM OCTOBER 2016

OVERVIEW

• What is inversion• Data or Observations• The model.• Model parameter (our target)• Forward Modeling• Inverse modeling

WHAT IS INVERSION?

Inversion is a common daily activity we usually apply to define, choose and take decisions. Such activity is usually practiced based on observations such as color, odor, density, …etc.

GENERAL EXAMPLES

Observations are needed to explore and differentiate between objects. Observations like size, color, density and even ornament can be used to determine the object.

MORE EXAMPLES

Stone egg (up) and true egg (down), how can we discriminate between them?

Answer:Density or weight, hardness, sound, heating.

We actually do inversion many times in our daily life. We use observations to choose the best fruits, vegetables or even used cars

We may call this descriptive inversion

DATA

Data is all the information and observations for our target problem, e.g. color, weight, luminosity, magnetic properties, …etc.

THE MODEL

• The model abstracts all our knowledge about the phenomena under investigation.

• It enables us to predict the observations of the phenomena using certain assumptions.

THE MODEL

• If you can reproduce the data by computation, then you have model and hence you can invert for model parameters (the target of inversion)

DEFINITION OF NOISEAny observable data that we cannot reproduce either numerically or analytical is called NOISE..

Hence what is considered as noise today, may become useful data tomorrow.

Example: ambient vibration seismic signals.

THE MODEL (TYPES)

• From the computational point of view

a. Analytical or exact.

b. Empirical models

c. Probabilistic or statistical models

ANALYTIC MODEL

• Models that are derived analytically, i.e. from mathematical physics. Produced through partial differential equations.

• Examples are wave equation and heat equation.

EMPIRICAL MODELS

Such kind of model depends on analyzing extensive observations of physical parameters. The model arise by finding the dependence between observations. This problem involves curve fitting of data. Some problems can not be solved via theoretical or analytic models may be solved via empirical model.

Example: Ohm Law

PROBABILISTIC MODEL

When our knowledge of a phenomena is not adequate, we rely on probabilistic model. Probabilistic model predict various output of the model then determine how likely this output will take place.

Example: Probabilistic earthquake assessment.

THE MODEL PARAMETERS

The model parameters is the objectives of the modeling. For example, the resistance in Ohm’s law may be regarded as the model parameter. In geophysics, model parameters may be velocity distribution in the subsurface.

MODEL MATHEMATICS

• Generally, Models can be represented in matrix form as:

d = G mWhere :d: is the data or observation vector.m: the model parameters vector.G: is the model kernel.

FINDING THE MODEL PARAMETERS

• Forward modeling

• Inverse modeling

FORWARD MODELINGAssumed

model parameters

Forward modeling

(computation of predicted observation

Comparison between

observations & prediction

Choosing different

model parameters

Computations exited when the camparison step yield measures less than the tolerance.

INVERSE MODELING

• In mathematical or matrix form means find the inverse of G such that the relation tends to be:

Mest = G-g d.Here G-g is called generalized inverse.

DATA

INVERSION

MODEL PARAMETER

S

APPRAISAL

GOODNESS OF MODEL

• Error misfit

• Resolution matrix

ERROR

Error measures or propagators is the misfit between the data and the calculated data. Despite of its well known acceptance, sometimes we have local minimum error with model parameters far from reality.

RESOLUTION MATRIX

Resolution matrix maps the estimated model with the true one. When the resolution matrix approaches Identity matrix, the estimated model is approaching the true model.

mest = G-g G mtrue+G-g ee is the error propagatorThe resolution matrix R = G-gG

INVERSION

Two types of Inversion

1- Linear Inversion

2- Non-linear invesion

LINEAR INVERSE PROBLEM

• Simple inversion in which the model parameters are not dependent on each other. The parameters is linearly mapped with observations.

• The common inversion technique for this class is the least squares.

LEAST SQUARES FIT

The problem here falls into three classes:1. Overdetermined (observations are more than the model parameter)2. Even determined (observations are equal to model parameters)3. Underdetermined (observations less than model parameters)

LEAST SQUARES

Class one is preferred one because it can give estimates for both model parameters and misfit. Underdetermined class, on the other hand, required information on model parameters from different experiment (analogous to non-uniqueness in linear Algebra)

LEAST SQUARES EXAMLES (OVERDETERMINED)

NON-LINEAR INVERSION

• A complex class of problems in which the mapping between the data and model parameters are not linear. The model parameters are interdependent and cannot be separated.

• Solution to this problem can be either through linearization if possible or iterative methods.

EXAMPLES

• Seismic tomography

• Location of earthquake hypocenters

NON-LINEAR INVERSE PROBLEM

• In seismic tomography the observed arrival time is not only dependent on the distribution of velocity in the subsurface but also on the location of earthquake hypocenter as well. The inversion is thus carried out for both.

Tomography example

SUMMARY• Inversion in general is one of our daily activity to identify, select and choose.

• Inverse problem is composed of three items, the data (observation), the model and the model parameters.

• Models are mathematical representation that enable the prediction of the phenomena (data).

• In determining the model parameters we can use either forward modeling or inverse modeling.

• Inverse problem can either be linear or non linear.

• Least squares curve fit is one of the common approaches to solve the linear inverse problems.

شكرا جزيالTERIMA KASIH

THANK YOU