talk on the inverse problem

1. BRIEF DISCUSSION ON
INVERSE THEORY
WEAKLY SEMINAR, PPSF@USM OCTOBER 2016

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

3. 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.

4. 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.

5. MORE EXAMPLES
Stone egg (up) and true egg (down), how can we
discriminate between them?
Answer:
Density or weight, hardness, sound, heating.

6. 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

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

8. 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.

9. 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)

10. DEFINITION OF NOISE
Any 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.

11. THE MODEL (TYPES)
• From the computational point of view
a. Analytical or exact.
b. Empirical models
c. Probabilistic or statistical models

12. ANALYTIC MODEL
• Models that are derived analytically, i.e. from mathematical physics. Produced through
partial differential equations.
• Examples are wave equation and heat equation.

13. 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

14. 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.

15. 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.

16. MODEL MATHEMATICS
• Generally, Models can be represented in matrix form
as:
d = G m
Where :
d: is the data or observation vector.
m: the model parameters vector.
G: is the model kernel.

17. FINDING THE MODEL PARAMETERS
• Forward modeling
• Inverse modeling

18. FORWARD MODELING
Assumed
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.

19. 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.

20. DATA
INVERSION
MODEL
PARAMETERS
APPRAISAL

21. GOODNESS OF MODEL
• Error misfit
• Resolution matrix

22. 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.

23. 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 e
e is the error propagator
The resolution matrix R = G-gG

24. INVERSION
Two types of Inversion
1- Linear Inversion
2- Non-linear invesion

25. 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.

26. 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)

27. 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)

28. LEAST SQUARES EXAMLES (OVERDETERMINED)

29. 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.

30. EXAMPLES
• Seismic tomography
• Location of earthquake hypocenters

31. 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.

32. Tomography example

33. 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.

34. ‫يال‬‫ز‬‫ج‬ ‫ا‬‫ر‬‫شك‬
TERIMA KASIH
THANK YOU