Signal Processing 1

The purpose of this chapter is to provide the fundamental concepts required to understand and analyze signals used in geophysics — such as seismic, well-logging, and electromagnetic signals.

We will describe their mathematical models, energy and power properties, orthogonal decompositions, and spectral (frequency) characteristics. 

Course Information

Course Objectives

The objective of this module is to provide Master’s students in applied geophysics with the mathematical and practical foundations of signal analysis, including the basics of deterministic and random signals, Fourier series and Fourier transform, convolution and correlation, and digital filtering.

Students will learn to interpret seismic signals in the frequency domain, apply advanced transforms such as 2D/3D Fourier, Hankel, Laplace and others, and understand their relevance in seismic imaging and inversion.

The course also develops practical skills through exercises and computational work, enabling students to implement spectral analysis and filtering techniques on synthetic and real geophysical data.

Prerequisites

A solid foundation in mathematics and physics is required for effective understanding of the course material.

Course Content

I / Review:

·       Integration of complex functions

·       Fourier Series and Fourier Transform, and their properties

·       Convolution

·       Correlation

·       Sampling

II / Digital Filtering

III / Various Other Transforms:

·       Two-dimensional Fourier Transform

·       Hankel Transform

·       Fourier Kernel

·       Three-dimensional Fourier Transform

·       N-dimensional Hankel Transform

·       Mellin Transform

·       Abel Transform

·       Fractional Fourier Transform

·       Laplace Transform

Tutorials (TD):
Practical exercises applying the course material to the different transforms.

Evaluation Method

Assessment is based on continuous evaluation and a final written examination.

References

• Oppenheim, A.V., Schafer, R.W. (2010). Discrete-Time Signal Processing. Prentice Hall, Upper Saddle River, 1120 p.
• Bracewell, R.N. (2000). The Fourier Transform and Its Applications. McGraw-Hill, New York, 640 p.
• Debnath, L., Bhatta, D. (2014). Integral Transforms and Their Applications. CRC Press, Boca Raton, 565 p.
• Yilmaz, Ö. (2001). Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data. Society of Exploration Geophysicists (SEG), Tulsa, 2027 p.