Geophysical Data Processing Theory

Code:

MEMG0006

Acronym:

TMG

Keywords
Classification	Keyword
OFICIAL	Mathematics, Physics, Earth Sciences

Instance: 2010/2011 - 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	9	Plano de estudos oficial a partir de 2008/09	1	-	6	56	162

Teaching language

Portuguese

Objectives

To familiarise students with 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 the treatment of signal in geophysics, namely seismic and gravimetry, as well as temporal series, for instance, climate data, seismic registers, etc…
This knowledge should be well consolidated, enabling students to deepen their knowledge, if their professional activity so demands it.

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 distribution. Integral of 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.
Stock questions. 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, cardinal seine, triangle, sampling function.
Convolution theorem. Parseval’s theorem or energy theorem.
Rayleigh’s theorem. Carleman theorem. Indefinite integral transform.
Moment theorem. First-order moment, centroid, moment of inertia, medium quadratic abscissa, variance, order n moment.
Relationship between signal compression and spectral expansion.
Spectral moment. Bernstein-Boas theorem.
Equivalent width. Sommerfeld theorem. Auto convolution. Autocorrelation width. Quadratic width. 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.
Concentrated constants systems. Wiener cybernetic method to approach the theory of systems. Linear time-invariant systems. Impulse response. Vaschy’s theorem. Harmonic response. Passage function. Filter design.
Linearity spectral meaning. Causality spectral meaning. Kramers-Kronig relationships. Hilbert transform. Phase, group and front delays.
Linear systems as media devices: proportional system, derivate systems. Integrator systems. Detection of seismic motions.
Seismograph differential equations; its reading in system passage function. Displacement, velocity and acceleration sensitivity.
Multivariable and field distributions. Multidimensional distributions: Dirac distribution families; convolution of distribution; derivation; divergence formula; gradient formula; Green theorem.
Gravimetry direct interpretation. Equivalent stratum theorem. Prolongation theorem. Direct interpretation by descending prolongation. Problem of filter synthesis.

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 essentially theoretical. 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, which are 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 treatment, as well as synthesis skills.

Software

Matlab 6

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	56,00
	Total:	-	0,00

Eligibility for exams

Students cannot miss more classes than allowed by the rules and have to reach a minimum grade of six out of 20 in the continuous assessment grade.
The assessment comprises: 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:
From 70% to 75% of the final grade: three tests
From 25% to 30% of the final grade: assignment and students’ performance.

Calculation formula of final grade

Final grade will be only based on the continuous assessment grade. Students have to attend a recurso (resit) exam to improve their grades. However, they have to be admitted to exams. (See Admission to Exams)

Examinations or Special Assignments

Not applicable

Special assessment (TE, DA, ...)

According to General Evaluation Rules of FEUP

Classification improvement

Recurso (resit) exam