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Data Processing Chapter 3 As a science major, you will all eventually have to deal with data. As a science major, you will all eventually have to deal with data. All data has noise All data has noise Devices do not give useful measurements; must convert data Devices do not give useful measurements; must convert data The better you can handle data, the more employable you will be The better you can handle data, the more employable you will be

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Wavelength and Fourier Analysis The granite produces a negative gravity anomaly The granite produces a negative gravity anomaly Variations in the sediment cover cause noise in the data Variations in the sediment cover cause noise in the data The noise and the anomaly have different wavelength scales The noise and the anomaly have different wavelength scales – Fourier analysis Separates signals by wavelength Separates signals by wavelength A Simple Example…

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General Wave Terms Amplitude (a) = Amplitude (a) = Wavelength (λ) = Wavelength (λ) = Frequency = Frequency = Period = Period = 0.5 λλ1.5 λ2 λ -a 0.5a 0 a Amplitude Wavelength (λ)

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Harmonic Analysis Harmonics: multiples of a signal’s half wavelength, L Harmonics: multiples of a signal’s half wavelength, L Why use harmonics? Why use harmonics? – Found in nature and music Like a guitar! Like a guitar! – Any wiggly line can be mathematically reproduced by adding together a series of waves Exact match requires ∞ waves Exact match requires ∞ waves 1 st 5 harmonics 0.2 L0.4 L0.6 L0.8 L L 0.5 0 1

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Fourier Analysis A type of harmonic analysis where wiggly data are separated into various harmonics of differing amplitude 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

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In this tide data, what type of wavelengths (short or long) should we remove? In this tide data, what type of wavelengths (short or long) should we remove? What causes each? What causes each? What to Remove?

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Caveats of Fourier Analysis Requires a complete signal Requires a complete signal – Starts and ends at same value Only analyzes wavelengths that are multiples of the signal length Only analyzes wavelengths that are multiples of the signal length Geologic targets likely have multiple wavelengths and may share some wavelengths with noise. Geologic targets likely have multiple wavelengths and may share some wavelengths with noise.

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Digital Filtering An alternative way to remove unwanted wavelengths/frequencies: Filtering An alternative way to remove unwanted wavelengths/frequencies: Filtering Usually applied to regularly space data Usually applied to regularly space data – If data not regular, interpolation can be used E.g. A simple 3-point filter :: 1/3 (y n-1 + y n + y n+1 ) E.g. A simple 3-point filter :: 1/3 (y n-1 + y n + y n+1 ) – Also called 3-point running average 3-point running average 3-point moving window 3-point moving window 1 0 5 4 9 4 2 3 1 5 1 2 3 4 5 6 7 8 9 10 Time or Distance Value Filtered Value 2X 36 There are also 5-point, 7-point, and n-point filters. There are also 5-point, 7-point, and n-point filters. – Some are weighted to remove certain wavelengths

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Effects of Digital Filters Low-pass filter Low-pass filter – Also called Smoothing filter – Allows low freq to “pass through” High-pass filter High-pass filter – Allows high freq To “pass through” Band-pass filter Band-pass filter – Constructed to only let certain “bands” or frequencies through A subwoofer in a bandpass box

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Effects of Digital Filters A given filter may have a very different effect on data depending on: A given filter may have a very different effect on data depending on: – Wavelength – Sampling Rate / Resolution A filter can completely decimate a signal A filter can completely decimate a signal

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

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Gridded Data Processing All of the data processing techniques discussed here can also be applied to gridded or even three-dimensional data All of the data processing techniques discussed here can also be applied to gridded or even three-dimensional data – Can filter directional noise – Highlights directional features – Geologists should pay attention in multivariable calculus class! – Geologists should pay attention in multivariable calculus class! E.g. MAT 2130 - CALCUL ANALY GEOM III E.g. MAT 2130 - CALCUL ANALY GEOM III – Do not rely on black box computer programs E.g. Arc GIS E.g. Arc GIS

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Filtering in 2D :: Gridded Data Filters can be created to filter all types of data Filters can be created to filter all types of data No technique is perfect No technique is perfect – Great care must be given when creating a filter or processing data in general

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