Seismic Signal Processing

Notebooks:
Signal Processing
Obspy
Deep Denoiser

Signal Processing 101

• Fourier Transform (FFT)
• Filtering
• Spectrogram
• Convolution and Cross-correlation
• Short-time Fourier Transform (STFT)
• Wavelet Transform
• Hilbert Transform
• ...

Fourier Transform

Fourier Transform (FT) is a mathematical operation that decomposes a function into its constituent frequencies.

The Fourier Transform of a function is given by:

The inverse Fourier Transform is given by:

Example: Fourier Transform of a Sine Function

Example: Fourier Transform of a Box Function

Wiki

Fourier Transform

Fourier Analysis

Filtering

Filtering is a process of removing unwanted components or features from a signal.

Example: Butterworth Low-pass Filter

Example: Chebyshev Filter

Convolution

Convolution is a mathematical operation on two functions and that produces a third function that expresses how the shape of one is modified by the other.

Convolution in Frequency Domain

Cross-correlation

Cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other.

Cross-correlation in Frequency Domain

Seismic Signal Processing using Obpsy

Deep Denoiser

• Short-time Fourier Transform (STFT) + Wiener Filter + Neural Network

Short-time Fourier Transform (STFT)

Time-frequency representation of a signal.

Wiener Filter

The Wiener filter minimizes the mean square error between the estimated signal and the desired one. When the signal and noise are stationary and have known power spectral densities, the Wiener filter can be expressed in the frequency domain as:

• is the frequency response of the Wiener filter.
• is the power spectral density of the desired signal.
• is the power spectral density of the noise.

Neural Network

Deep Denoiser for Seismic Data

Paper