Study-Unit Description

Study-Unit Description


TITLE Time Series Analysis, DSP and Inverse Theory

LEVEL 03 - Years 2, 3, 4 in Modular Undergraduate Course


DEPARTMENT Geosciences

DESCRIPTION We live in a digital age in which the collection and storage of data happens continuously and in real-time. Large bases of data that store the recorded parameters are continuously being created and populated with measurements. However, only through the understanding of the underlying concepts and intelligent processing can new and valuable knowledge be extracted.

In this study-unit, processing techniques that are frequently applied to map the datasets onto different domains, extract hidden signals and enhance the signal-to-noise ratio, are put forward.

The mathematical background and application of specialised algorithms that are used to extract signals from noisy measurements or to identify new hidden signals from complex dataset, will be presented.

Students following this study-unit will become equipped with an important tool-set that is applicable to all fields of Geoscience.

Study-unit content:

• Introduction: Analog vs digital; finite vs infinite; linear time invariant systems; dimensionality; Moving averages, smoothing and exponential smoothing techniques error metrics;
• Convolution and correlation operators;
• Fourier Series: decomposition of periodic functions (continuous, finite time series), spectral analysis, windowing;
• Continuous Fourier Transform: definition and properties; convolution and deconvolution; Circular convolution;
• Discrete Fourier Transform: definition and properties, Fast Fourier Transforms (FFT), sampling issues (aliasing, Nyquist, etc.); Discrete Fourier Transform for long data sequences;
• Laplace Transform: definition and properties; relevance to differential equations; transfer functions; discrete version (Z-transform);
• Correlation and covariance; power spectra;
• Principal and independence component analysis (PCA and ICA);
• Digital Filters and their applications: low-pass, high-pass, bandpass, notch, Butterworth, poles and zeros, time domain vs frequency domain; Design of digital filters; wavelets and applications as filters, f-k filtering;
• Introduction to Inverse Theory: Useful Definitions; forward vs inverse problems; goals of an Inverse Analysis; Examples of Forward Problems;
• Inverse methods based on length: Data Error and Model Parameter Vectors; Minimizing the Misfit: Least Squares; Determinacy of Least Squares Problems; Minimum Length Solution; Weighted Measures of Length; Variance of the Model Parameters;
• Linearization of nonlinear problems: General Procedure for Nonlinear Problems and examples;
• The eigenvalue problem: Eigenvalue Problem for Square for a square matrix; Geometrical Interpretation of the Eigenvalue Problem; Decomposition Theorem for Square matrix; Eigenvector Structure; singular-value decomposition;
• Non-linear inverse problems: local techniques (gradient and curvature based iterative methods); global methods (grid search and genetic algorithms).

Study-unit Aims:

The aim of this study-unit is to introduce the students to the theories behind the processes that are routinely applied to field measurements before new and useful information can be extracted. Following the mathematical definitions, each method is demonstrated through a real application on collected data.

The study-unit aims to:
- introduce the fundamental concepts and methods that are used to analyze, manipulate and extract information from digital signals;
- make use of simple programming techniques as well as available software packages to instil in students a familiarity and confidence with working with digital time series, which are the basis of much of data analysis in the geosciences;
- familiarise students with the basic theory behind inversion methods;
- introduce available methods, and their limitations, for extracting information about the earth from available data sets, using modern methods and computer programs for mathematical inversion.

Learning Outcomes:

1. Knowledge & Understanding
By the end of the study-unit the student will be able to:

- analyse time series for deterministic or statistical behaviour; represent time series with linear functions, polynomials or splines;
- quantitatively analyze the errors in datasets;
- carry out sampling and reconstruction of continuous time signals, characterization and properties of discrete time signals and systems;
- carry out operation involving the representation, manipulation and processing of time series in both time and frequency domains;
- explain in a quantitative manner the principles behind the convolution and the correlation operators on multi-dimensional datasets;
- separate signals into different components to extract and identify any underlying periodicity;
- compute and apply the discrete time Fourier transform and Laplace transform (Z-Transform);
- select the optimum methods and parameters for spectral analysis of time series and the extraction of information from them;
- recall the basic principles of digital filter design techniques;
- explain the processes leading to aliasing and spectral leakage and design filters to prevent aliasing and leakage;
- appreciate and apply basic linear inverse theory;
- implement a variety of techniques for solving linear and non-linear problems.

2. Skills
By the end of the study-unit the student will be able to:

- load and visualise a measured time series in Matlab;
- apply spatial and frequency filters to the data;
- compute error metrics between modeled and observed data;
- apply univariate and multivariate models;
- compute correlation, autocorrelation and cross-correlation;
- apply Fourier and Wavelet transforms;
- perform component analysis;
- apply basic models to solve simple inverse problems;
- build on their programming skills and enhance their mathematical background.

Main Text/s and any supplementary readings:

- Ifeachor, Emmanuel C. and Jervis, Berrie W. (2001). Digital Signal Processing: A Practical Approach. Upper Saddle River, NJ: Prentice Hall.
- Press, William H. and Teukolsky, Saul A. and Vetterling, William T. and Flannery, P. (1992). Numerical Recipes in C: the art of scientific computing. Cambridge University Press.
- Parker R. (1994), Geophysical Inverse theory, Princeton University Press.
- Gubbins, D. (2010) Time series Analysis and Inverse Theory for geophysicists, Cambridge University Press.
- Kanasewich, E. (1981) Time Sequence analysis in Geophysics, University of Alberta Press.

ADDITIONAL NOTES Pre-Requisite Study-unit: GSC2400

STUDY-UNIT TYPE Lecture and Tutorial

Assessment Component/s Assessment Due Resit Availability Weighting
Assignment SEM1 Yes 40%
Examination (1 Hour and 30 Minutes) SEM1 Yes 60%

LECTURER/S Sebastiano D'Amico
Adam Gauci

The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints.
Units not attracting a sufficient number of registrations may be withdrawn without notice.
It should be noted that all the information in the description above applies to study-units available during the academic year 2021/2. It may be subject to change in subsequent years.