This chapter provides the fundamentals of optimum filtering and shows different applications of this important theory in seismic data processing.

An introduction to motivate readers regarding the subject of seismic signal processing. It also focuses on general seismic data acquisition, processing workflow, the seismic convolution model and seismic interpretation.

Seismic deconvolution is at the heart of seismic data processing. Deconvolution can be done determinsitically, via optimum filtering in time or in other domians. This chapter discusses the principles of seismic deconvolution and shows various techniques with examples.

This chapter presents sampling theorem for seismic data, including Shannon sampling theory for sampling continuous time (space) signals. Also, we explain the aliasing effects due to under-sampling of seismic data sets. Moreover, the theory of compressive sensing (CS) is currently considered the state-of-the-art theory of DSP, with many applications related to signal and image compression, signal recovery, and many other applications. CS is currently used for various seismic data processing problems. Hence, in this chapter we introduce CS principles and provide a few seismic data processing-related applications.

Seismic applications of digital filtering theory are presented in this chapter. 1-D FIR and/or IIR digital filters, such as low-pass or band-pass, are used heavily to enhance the signal-to-noise (SNR) ratio of acquired seismic data. Furthermore, 2-D digital filters like fan filters have become standard in removal of surface waves accompanying seismic data records. Solving the wave equation numerically may also require using FIR or IIR digital filters such as the explicit depth wavefield extrapolation filters.

Seismic wavelets model so many signals, including seismic source signatures, and are a main part of the seismic convolution model. They can be classified in various types. This chapter discusses various types of wavelet and their importantce. Also, it presents seismic wavelet processing as a method to shape the seismic wavelet, i.e., reduce its effect on seismic data sets.

An intensive overview of the fundamentals and physical principles on which seismic methods are based. It provides the necessary related geophysical background to understand seismic data and, hence, the reader will obtain a more clear understanding of how to properly process the data in order to ultimately obtain better seismic images that are used for accurate interpretation.With various examples, this includes the theory of elasticity, the wave equation, the types of seismic waves, single-layer reflector models, seismic events, etc.

Useful discrete-time signals and systems properties are introduced. This is followed by a brief review of the z-transform.Spectral analysis of seismic data and useful transforms are discussed. Signal analysis in the spectral or other domains is very important and assists in obtaining a better understanding of signals. Particularly when dealing with seismic data, it becomes almost standard to analyze seismic data sets in the 2-D frequency-wavenumber domain. Also, other discrete transforms such as the Radon transform are very useful for processing seismic data sets, which can be used, for example, for seismic wavefield decomposition as well as seismic multiple removal.

In addition, ρ is the density, K is the bulk modulus, µ is the shear modulus, λ is Lamé’s coefficient, E is Young’s modulus, and v is Poisson’s ratio.