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Geol 335.3 LAB 6: Fourier transforms and Filtering |
This lab has two parts dealing with Fourier analysis, and the relations between the time and frequency domains.
This exercise is designed to familiarize you with Fourier transforms. You will need to make sure you are using a browser with Java enabled - you can run this part of the lab on any system. If you are using Netscape on a Unix box, look under "Options" for "Network preferences", then click on "Languages" and make sure the box for Java is checked.
Use the Fourier transform tool :
Set the checkbox “origin centered”. Zero all the coefficients. The top two panels are the real and imaginary parts of the time domain. The bottom two panels are the real and imaginary parts of the signal in frequency domain.
Zero time is shown as the open circle in the top panels, zero frequency is shown as the open circle in the bottom. Make plots of your work. To plot, it is usually sufficient to click File->Print in your browser.
1) (5%) Put in a “boxcar” function (square wave) 5 units wide (a box of 5 samples of equal values) centered on the origin. What is the Fourier transform of this? How wide is the main lobe? Print the plot out and mark it up.
2) (5%) Double the width of the square wave function. How wide is the transform? What property of the Fourier transform does this illustrate?
3) (5%) Move the boxcar function so that its rightmost sample is at the origin. What does the transform look like? What property of the Fourier transform does this illustrate?
4) (5%) Put a single impulse at the origin. What does its transform look like?
5) (5%) Move the impulse 2-4 time samples away from the zero grid point. What happens to the transform?
6) (5%) Put in two impulses equally spaced around the origin two units away from the origin. This is called an even impulse pair. What is the transform?
7) (5%) Make one of the impulses negative. This is called the odd impulse pair. What does the transform look like?
8) (5%) Put the even impulse pair two units away from the origin on the real part of the frequency domain. What is the time domain signal look like? What property of the Fourier transform do questions 6 and 8 together illustrate?
9) (5%) Perform step (8) using an odd impulse pair, but put it on the imaginary part of the Fourier transform. What does the transform look like?
10) (5%) In the origin centered mode, the first sample in the frequency domain is the Nyquist frequency. Put a single spike there. What is the time domain signal?
Play around with the transform tool.
The most common use of bandpass filtering on real data is to try to remove noise when the signal and noise don’t overlap too much in frequency. In this exercise, you will use the seismic processing package ProMAX to look at a shot gather with a (hopefully) obvious noise problem, and remove it using a bandpass filter.
Start ProMAX by typing promax in a command shell on your Linux workstation. In the main ProMAX window, select your Area and Line (they should be prepared for you for these labs). The “Line” should have a sample dataset and a geometry database associated with it.
When you click on a Line name, you will be taken to the list of the processing flows associated with the line. Most likely, the list just includes a single flow used to create the geometry. Create a new processing flow named "Display”.
1) (5%) The sampling interval of this data is 2000 ms = 2 ms. What is the Nyquist frequency for this dataset?
After you create the flow, you are taken to the flow builder screen. On its right, you have a list of the available seismic processing tools; on the left is the sequence of tools you select for processing. To perform this task, you will need the following tools: Disk Data Input, Bandpass Filter, and Trace Display. Note that the tool sequence is important – the tools lower in the flow receive their inputs from those above them. Use a frequency band of 4-8-24-48 in the filter.
Execute the flow (find the corresponding button above the tool list). The Trace Display will pop up a window of shot gathers. Scroll to a shot you like. Use the zoom buttons (on the left in the Trace Display toolbar) to pick an appropriate zoom.
2) (20%) Make a hardcopy plot of the data (use File->Hardcopy in the Trace Display), label the main phases, and sketch the areas that you think are mostly signal and those where there is a lot of noise. How did you decide between them?
3) (5%) What periods/frequencies can you see in the noisy area?
Now you will test 2 series of filters 1-octave slopes at each end and a 1-octave bandpass (i.e. four numbers with frequency doubling each time: 2-4-8-16 for example). The two series need to overlap: often a series such as
2-4-8-16
3-6-12-24
4-8-16-32
6-12-24-48
8-16-32-64
12-24-48-96
You need to choose series that start below the lowest frequencies you can see in the data (typically seismic sources don't have much energy below about 6Hz), and extend up to the Nyquist frequency. This means that we can choose corner frequencies to within half an octave anywhere in the frequency range of our data.
To apply the filters, exit the flow builder and copy the Display flow into a new flow named "Filter Panels". In the new flow, add "Parameter Test" tool before the "Bandpass Filter". In the Parameter Test, give the list of parameters that you would like to pass to the Filter, and in the Filter, give "99999" as the value of the filter parameter string. This code will mean that the parameters would be obtained from Parameter Test.
In Trace Display tool, find the setting to make the number of shot gathers displayed at a time equal the number of tested filter parameter groups plus one. This would simplify viewing and comparison of the resulting filter panels.
4) (0%) Make hardcopies of the resulting panels - this is a set of filter test panels.
5) (5%) Which panels are mostly noise and which are mostly signal?
6) (10%) Now mix the filter panel ranges to choose a single bandpass filter that includes the range of the signal but excludes the noise. What are the four corner frequencies? Note that lower-frequency contains information about the deeper reflections yet more ground roll - you have to balance between these two criteria.
7) (5%) Make a hardcopy plot of the data with this bandpass filter and compare to the original and to the filter panels. Did this bandpass remove anything you think was significant in the data?
Answers to the questions and the plots in a binder.