Geophysical Data Processing Theory

Code:

MEMG0006

Acronym:

TMG

Keywords
Classification	Keyword
OFICIAL	Mathematics, Physics, Earth Sciences

Instance: 2012/2013 - 2S

Active?	Yes
Web Page:	http://moodle.fe.up.pt
Responsible unit:	Mining Engineering Department
Course/CS Responsible:	Master in Mining and Geo-Environmental Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MEMG	14	Plano de estudos oficial a partir de 2008/09	1	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

To familiarise students with the Fourier Transform, supported by the previous teaching of the theory of tempered distributions. Illustration of the theory with applications in the domain of mining engineering and geoenvironment, namely geophysics. Application of the theory on signal processing in geophysics, particularly in seismic. This knowledge should be well consolidated, enabling students to deepen their knowledge, if their professional activity so demands it.

Learning outcomes and competences

 

In completing this course, basically the student should be able to:

 

1. Conveniently apply the sampling theorem in A/D operations.
2. Conveniently apply the direct and inverse Fourier transform.       
3. Understand the application of filters in signal processing in the time7space and frequency domains
4. Develop data processing algorithms in Matlab environment.

 

Working method

Presencial

Program

Concept of distribution. Examples of application of the concept. Properties of distributions. Distribution as a generalized function. Properties of tempered distributions. Convolution. Identification of a distribution with an interval function. Identity of two distributions in an interval. Limited support distribution. Derivative of a distribution. Odd and even distributions. Integral of a distribution. Properties of Dirac distribution. Distribution limits. Distributions as function limits. Example using pedestal and Gaussian function. Physical quantities as distributions – impulse response Convolution: definition, commutative, associative property, etc Translation as convolution. Derivation as convolution. Integration as convolution. Z transform. Deconvolution. Fourier transform. Definition. Euler relation. Cissoid properties. Presentation of Fourier transform. Fourier theorem (inversion formula). Sine and cosine transform. Transform representation in module and phase. Demonstration of Fourier theorem. Fourier integral of a real function. Fourier integral of a pure imaginary. Fourier integral of a real and even function. Fourier integral of a real and odd function. Fourier integral of any real function. Fourier integral of a hermitian function. Fourier integral of a causal function – interdependence of a real part and imaginary spectrum. Physical meaning of Fourier transform: spectrum. Amplitude spectrum, phase spectrum, spectral density, properties of Fourier operator. Alternative definitions. Spectrum elementary properties: Linearity, symmetry, translation, modulation, scale, derivation, integration. Spectrum of some functions and interesting distributions: Dirac distribution spectrum, sinusoid, Heaviside step, pedestal, sinus cardinal, triangle, sampling function. Convolution theorem. Parseval’s theorem or energy theorem. Rayleigh’s theorem. Relationship between signal compression and spectral expansion. Heisenberg theorem or uncertainty principle. Functions and spectrum of limited support. Flatness. Gibbs phenomenon. Periodic functions and Fourier series. Shannon and Kotielnikov theorem. Gabor theorem. Current and instantaneous spectrum, spectral density. Linear time-invariant systems. Impulse response. Passage function. Introduction to filter design.

Mandatory literature

Papoulis, Athanasios; The Fourier integral and its applications. ISBN: 07-048447-3
Karl, John H.; An introduction to digital signal processing. ISBN: 0-12-398420-3
Novais Madureira - Abilio Cavalheiro; Teoria dos Métodos Geofísicos, 2008

Complementary Bibliography

Bracewell, Ronald N.; The Fourier transform and its applications. ISBN: 0-07-007013-X

Teaching methods and learning activities

This course unit is theoretical-practical. The theoretical-practical examples aim to consolidate students’ knowledge. Furthermore the theoretical-practical examples show the constrictions imposed by discrete sampling in a small interval (non-infinitesimal) of numerical examples, in contrast with the continuous sampling from minus infinity to plus infinity with infinitesimal step which corresponds to the analytical formulation. At a more advanced stage, all theory converges to the understanding of sampling methodologies and signal processing, as well as synthesis skills based on transfer functions describing linear systems.

Software

Matlab 6

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	63,00	80,00
Mid-term exams	Teste	5,00	20,00
	Total:	-	100,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
Attendance	Frequência das aulas	63	2013-06-07
Study	Estudo autónomo	126
	Total:	189,00

Eligibility for exams

Attended classes in number equal or exceeding the minimum allowed by FEUP General Evaluation Rules and a minimum final grade of six out of 20, in the students continuous assessment. The assessment comprises: 2 tests, assignments and students’ performance. The grade is based on the average grade of all components. Students with special status have to perform the same practical assignments and attend the same tests as regular students. Continuous assessment grade: 65% of the final grade for the two tests and 35% for assignments and students’ performance.

 

Calculation formula of final grade

Final grade will be the continuous assessment grade. If required students may attend a recurso (resit) exam aiming at improving their grades. However, they have to be admitted to exams. (See Obtenção de frequência)

Examinations or Special Assignments

Not applicable

Special assessment (TE, DA, ...)

According to General Evaluation Rules of FEUP

Classification improvement

Recurso (resit) exam