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Data Processing and Fourier Analysis: Handling Noise and Separating Signals, Slides of Geology

Chennai Mathematical InstituteGeology

An introduction to data processing, focusing on the challenges of dealing with noisy data and the use of fourier analysis for separating signals by wavelength. It covers the concepts of harmonics, wavelength terms, and digital filtering, as well as the effects of digital filters and the concept of aliasing.

Typology: Slides

2012/2013

Uploaded on 07/19/2013

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Data Processing
Chapter 3
•As a science major, you will all eventually have to deal with
data.
•All data has noise
•Devices do not give useful measurements; must convert data
•The better you can handle data, the more employable you will be
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Download Data Processing and Fourier Analysis: Handling Noise and Separating Signals and more Slides Geology in PDF only on Docsity!

Data Processing

Chapter 3

  • As a science major, you will all eventually have to deal with

data.

  • All data has noise
  • Devices do not give useful measurements; must convert data
  • The better you can handle data, the more employable you will be

Wavelength and Fourier Analysis

  • The granite produces a negative gravity anomaly
  • Variations in the sediment cover cause noise in the data
  • The noise and the anomaly have different wavelength scales - Fourier analysis - Separates signals by wavelength

A Simple Example…

Harmonic Analysis

  • Harmonics: multiples of a signal’s half wavelength, L
  • Why use harmonics?
    • Found in nature and music
      • Like a guitar!
    • Any wiggly line can be mathematically reproduced by adding together a series of waves - Exact match requires ∞ waves

1 st^ 5 harmonics

0.2 L 0.4 L 0.6 L 0.8 L L

-

0.

0

0.

1

 

  

=  L

y a sin^ π nx

Fourier Analysis

  • A type of harmonic analysis

where wiggly data are separated into various harmonics of differing amplitude

  • Adjusts the amplitudes of each harmonic
  • Can isolate dominant frequencies/wavelengths in data and remove unwanted ones
  • Sum of ∞ harmonics reproduces data exactly

Sum of same harmonics, but with different amplitudes

Data Separated by Wavelength

Caveats of Fourier

Analysis

  • Requires a complete signal
    • Starts and ends at same value
  • Only analyzes wavelengths

that are multiples of the signal length

  • Geologic targets likely have

multiple wavelengths and may share some wavelengths with noise.

Digital Filtering

  • An alternative way to remove unwanted

wavelengths/frequencies: Filtering

  • Usually applied to regularly space data
    • If data not regular, interpolation can be used
  • E.g. A simple 3-point filter :: 1/3 (yn-1 + yn + yn+1)
    • Also called
      • 3-point running average
      • 3-point moving window

Time or Distance 1 2 3 4 5 6 7 8 9 10 Value Filtered Value (^) X 2 3 6

  • There are also 5-point, 7-point, and n-point filters.
    • Some are weighted to remove certain wavelengths

Effects of Digital Filters

  • A given filter may

have a very different effect on data depending on:

  • Wavelength
  • Sampling Rate / Resolution
  • A filter can

completely decimate a signal

Aliasing

  • If sampling rate (resolution) approaches wavelength of signal - May see false patterns
  • If sampling rate is less than signal’s wavelength - May see false long wavelength signals
  • Aliasing : Discrete (non- continuous) data can suggest patterns that are not real
  • Nyquist wavelength = half the signal’s wavelength. - This is the minimum sampling rate to avoid aliasing

Filtering in 2D :: Gridded Data

  • Filters can be created to filter all types of data
  • No technique is perfect
    • Great care must be given when creating a filter or processing data in general