## Practical Mathematics and its Applications in Seismic Processing

John Bancroft

This course reviews mathematics from high-school and university, and then expands their concepts with practical examples to provide the fundamentals that are required for developing practical applications and algorithms in seismic processing.

The content of the course is designed to include practical examples from seismic processing to reinforce the basic concepts, and will be repeated, where appropriate, to maintain a common thread throughout the development of the mathematical material.

Two days

### Intended Audience

Entry and Intermediate Level

### Prerequisites (Knowledge/Experience/Education Required)

It is assumed the participants have had some mathematics beyond the high-school level, and pose a happy, eager, and searching mind.

### Course Description

The course will commence with a review of the basic mathematics (math) that was studied in high school and possibly at university including calculus, complex numbers power series and linear algebra. These topics are applied with a review of signal processing examples. A foundation of systems follows with topics such as superposition, modelling, and inversion. Power series and transforms (Fourier, Radon, “z”) are followed by numerical methods and linear algebra.

Throughout the course, extensive use of examples will be given. Most of the material will be presented with extensive figures that relate theoretical concepts with practical applications.

Seismic application include field design, noise attenuation, filtering, stacking, seismic deconvolution, velocity analysis, surface consistent processing for statics, amplitudes, and deconvolution, inversion, tomography, wavefield modelling, migration, least squares migration, and full waveform inversion.

### Learner Outcomes

The participants will know the fundamentals of geophysical mathematics to enable them to read scientific articles and apply this information to building seismic applications that are understandable, accurate, and efficient.

• Comprehend and understand the mathematical algorithms used in geophysics.
• Write and implement the programming code in accurate and efficient programs.
• Identify and improve weakness in current programming code.
• Know why mathematical procedures are performed in specific orders.

John Bancroft