Lecturers:
Dr Steve Hearn
Course Summary:
This course provides a sound understanding of the handling of physical data in digital form. It instills general concepts and philosophies, as well as introducing specific techniques relevant to geophysics. The course is biased towards seismic analysis, since that is where much of the material originated. Nevertheless the concepts are generally applicable throughout geophysics.
The course examines general principles of digital filtering, including digitisation concepts and representation in different domains (time, frequency, Z-transform). The concept of wavelets is used to examine phase characteristics. Exact and least-squares inverse filtering are discussed. Specific seismic applications (deconvolution, layered-media transmission) are used as illustrative examples.
The course includes a number of programming assignments which are closely integrated with the lecture material. These assignments assist in the understanding of concepts. In addition, they provide a sound introduction to practical scientific computation.
Contact:
2 hours lectures per week plus 1 hour tutorial
Lecture Room:
Exploration Geophysics Laboratory - Steele Building
Assumed Background:
ERTH4120 is normally taken during the final year of a professional geophysical qualification, although it can also be done as an elective for other disciplines such as Engineering, Physics, Geology or Mathematics. It assumes intermediate level background in mathematics (emphasis on applied and numerical). Prior exposure to geophysics and scientific computation are desirable though not essential. Contact Dr. Hearn for further details.
References:
Robinson,E.A. and Treitel, S. (1980) Geophysical Signal Analysis, Prentice Hall, 466p.
Robinson, E.A. (1983) Multichannel Time Series Analysis. Goose Pond Press, 455p.
Hatton, L. , Worthington, M.H., Makin, J. , Seismic Data Processing, Blackwell.
Press, W.H., Teukolsky,S.A., Vetterling,W.T. and Flannery,B.P. (1992) Numerical Recipes in Fortran. The Art of Scientific Computing. 2nd Edition, Cambridge University Press, 933p.
Yilmaz, O., Seismic Data Processing, Society of Exploration Geophysicists.
Course Outline:
Digital Filtering
Analog to Digital Conversion and the sampling theorem
Causal digital filters
Constant filter, unit-delay filter
Series and parallel combinations, mixers
General nth order Causal Feedforward filter
Z-transform of a filter and of a signal
Convolution in time domain and via Z-transforms
Frequency-domain interpretation of digital filters
Transfer functions, magnitude and phase-lag spectra
Minimum and maximum phase-lag spectra
Relationship between Z-transform and Discrete Fourier transform
Relationships between time signal, complex spectrum, autocorrelation, power spectrum
Wavelets
Definitions of wavelets and time series
Partial energy concept
Autocorrelation and crosscorrelation of wavelets
Delay properties of wavelets
Delay properties in terms of partial energy
Delay properties in terms of roots of polynomial
Spectral factorisation – root method and Kolmogorov method
Inverse Digital Filters
Purpose and definition
Causal inverse of 2-length wavelets
Stability of causal inverse and relationship to delay properties
Memory functions
Anticipation functions and non-causal inverse of 2-length wavelets
Exact inverse of wavelet of length greater than 2
Practical problems in computation of exact inverses
Feedback filters and Autoregression
Simple feedback filters
Stability versus delay
Deghosting, dereverberation of water-bottom multiples
The three basic data models and corresponding spectral techniques
Wiener Filtering
Background concepts - energy and power signals, expectation operator
General filter design model, Wiener (least squares) criterion
Development of normal equations
Practical construction of Normal Equations from auto- and cross-correlations
Filter Performance Parameter
Factors influencing filter performance
Relationship of Wiener spiking filter to exact inverse filter
Design of optimum Wiener wavelet filters
Practical considerations for Wiener waveshaping
Predictive deconvolution
Random reflectivity assumption for predictive deconvolution
Z-Transform Approach to Transmission in Layered Media
Wave propagation across a boundary
Reflection and Transmission coefficients
Normal incidence transmission in a multilayered system
Scattering matrices
Reflection seismogram for simple 2 and 3 layered systems
Reflection seismogram for general multilayered systems
Extension to oblique incidence
Assessment:
Final Examination 75%, Assignments 25%
Examination material and assignment work is assessed on the technical accuracy of submitted work, scientific logic, and presentation. Further details on assessment procedures are provided prior to each assessable task.
Except where group work is specified, all work must be that of the author. Departmental policy on non-compliance with assessment procedures can be obtained from the Department of Earth Sciences office in the Steele Building.
Assignments:
A number of programming assignments are given to assist in understanding of lecture material, and also to develop sound practical programming skills. It is preferable, though not essential, that students have some previous experience in a high-level language such as Fortran, C, C++, Pascal or Java.
The assignment topics generally cover:
Individual handouts detailing approach and requirements are provided for each assignment.
Feedback on this assignment material is given throughout the semester. Students must return all assignment material for final assessment at the time of examination.