This chapter presents the fundamentals of the convolution operation and the related operation of correlation.

The theory and practice of convolutional neural networks (CNNs) are introduced where the dense matrices of a fully connected neural network (FCNN) are replaced by the sparse correlation matrices.

This chapter discusses how the convolutional sparse coding method can be used to eliminate both coherent and random noise in seismic data.

Abstract The most basic and commonly used one-dimensional model for the seismic trace is referred to as the convolutional model, which states that the seismic trace is simply the convolution of the earth's reflectivity with a seismic source function with the addition of a noise component. In equation form, where * implies convolution, An even simpler assumption is to consider the noise component to be zero, in which case the seismic trace is simply the convolution of a seismic wavelet with the earth’s reflectivity, In seismic processing we deal exclusively with digital data, that is, data sampled at a constant time interval. If we consider the relectivity to consist of a reflection coefficient at each time sample (some of which can be zero), and the wavelet to be a smooth function in time, convolution can be thought of as “replacing” each reflection. coefficient with a scaled version of the wavelet and summing the result. The result of this process is illustrated in Figures 2.1 and 2.2 for both a “sparse“ and a “dense” set of reflection coefficients. Notice that convolution with the wavelet tends to "smear" the reflection coefficients. That is, there is a total loss of resolution, which is the ability to resolve closely spaced reflectors. An alternate, but equivalent, way of looking at the seismic trace is in the frequency domain. If we take the Fourier transform of the previous equation, we may write In the above equation we see that convolution becomes multiplication in the frequency domain Howeve,

Abstract This paper introduces convolutional quelling as a technique to improve imaging of seismic tomography data. We show the result amounts to a special type of damped, weighted, least-squares solution. This insight allows us to implement the technique in a practical manner using a sparse matrix, conjugate gradient equation solver. We applied the algorithm to synthetic data using an eight nearest-neighbor smoothing filter for the quelling The results were found to be superior to a simple least-squares solution because convolutional quelling suppresses side bands in the resolving function that lead to imaging artifacts.