# EOSC 354 · Analysis of Time Series and Inverse Theory for Earth Scientists

Continuous and discrete Fourier transforms, correlation and convolution, spectral estimates, optimum least-squares filters, deconvolution and prediction, frequency-wave number filtering. A practical course on computer techniques applied to the analysis of a wide range of geophysical phenomena. [3-2-0]
Prerequisite: Either (a) SCIE 001 or (b) one of MATH 101, MATH 103, MATH 105, MATH 121 and one of PHYS 101, PHYS 107, PHYS 117, PHYS 157.

Michael Bostock

#### Textbook

Signal Processing and Linear Systems by B.F. Lathi Berkeley-Cambridge

#### Course Content

http://www.eos.ubc.ca/courses/eosc354/index.html

#### Lecture Topics

Week Topic
1

Complex numbers and decomposition by means of orthogonal functions. Chapters: B and 3

Comments: First, the beauty, and therefore the simplicity, of complex (simplex) numbers. Then, orthogonality, a central theme in all of data processing, analysis and inversion. Orthogonal systems, to put it simply, simplify life.

2

Fourier series and Fourier transform. Chapters: 1,3 and 4

Comments: The basic transform in ALL data processing. We begin with Fourier series and develop the Fourier transform (our first orhtogonal basis).

3

Special functions. Fourier transform theorems. Chapters: 1 and 4

Comments: Life is considerably simplified by the use of some ubiquitous functions and theorems. Simple functions, simple theorems. Fourier series and develop the Fourier transform (our first orhtogonal basis).

4

Linear systems. Causality and realizability. Chapters: 1,3 and 4

Comments: This course is ONLY about linear systems. Of course, life is non-linear, but first things first.

5

Sampling theorem, aliasing, reconstruction. Chapter: 5

Comments: Nature provides us with continuous data (please do not introduce quantum theory here), data that we must analyze digitally. How to sample and what the effects of this sampling are, is the subject here. first.

6

Discrete Fourier transform, FFT, circular and periodic convolution. Chapters: 8,9 and 10

Comments: Now that we are in the discrete domain, we have to learn how to use it and what the consequences are.

7

Z Transform. Chapter: 11

Comments: My love affair with signal processing began with the z transform. It is to the discrete what the ò (…) is to the continuous.

8

Dipole decomposition and minimum phase. Chapter: 9

Comments: This is mostly me + references to published jewels. The concept of minimum phase originated in electrical engineering with respect to stability and I have no idea why it is so hidden in our book. It does, of course, appear, pages 603-609, but we need to do better.

9

FIR and IIR filters. Chapters: 9,10 and 12

Comments: Filters with finite and infinite impulse response. This subject is scattered throughout the book and I will attempt to synthesize this important topic. Important because these filter go under different names in other disciplines (MA, AR and ARMA).

10

Recursive filters and seismic applications. Chapter: 11

Comments: Recursion is fundamental in data processing and in system modeling. The seismic applications will be presented with reference to published work.

11

Optimum Wiener filters. Chapter: None

Comments: Lathi has explored this topic, beautifully, for continuous systems in other books. Here, we need a little linear algebra and least squares, both of which I will introduce simply and cogently (I hope). Wiener received the Nobel prize for this work. He merits an introduction, at least.

12