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Geol 335.3 LAB 9: 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 seismic 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.
Login into a Unix session on computer named 'oka.usask.ca'. If you sit in
front of this computer, simply open a 'terminal' program. When sitting in front
of another Linux machine anywhere else on campus, you can probably use it as
well, or you can login to 'oka' by typing 'ssh -Y oka'. If using a Windows computer
, you can use the 'mobaexterm' software at:
https://mobaxterm.mobatek.net/
The "Portable edition" of this software does not seem to require installation. Use 'ssh' and 'X11' options to connect to 'oka.usask.ca' with interactive graphics. You should see the Unix "shell" (window). Type 'p5000' to start ProMAX, You should then see the main ProMAX window (interface called SeisSpace).
To begin, create an 'Project' (this is also called 'Area' in ProMAX) for yourself. Name this area conveniently using your name.
In the navigation bar on the left, select project 'Geol 483/335' and subproject 'Geol335 lab 9'. Copy this project to your area by using the commands in the 'File' menu. Do not attempt working further in the common class area.
Select your copy of 'Geol335 lab 9'. Below this level, expand the 'Flows' folder and select Click on flow 'Trace display'. Look at the sequence of tools used (only 'Disk data input' and 'Trace display'. When pointing at each tool, click middle mouse button and see the parameters used by these tools. Note that if you press on the right mouse button, the corresponding tool becomes activated or de-activated.
Execute the job by pressing 'Ctrl-L' (this can also be done from the menu on top). You will see that the job is submitted in a parallel process, and after a couple seconds, you should see the seismic trace display. By pressing arrows < and > in the upper-hand corner, you can scroll through the shot records.
The sampling interval of these data is 2000 ms = 2 ms (it can be found in the log files after running any flows). Knowing this, answer the following question:
1) (5%) What is the Nyquist frequency for this dataset?
Select another flow 'Shot 163'. This flow only loads and displays one shot from the same dataset. Look at parameters of the 'Disk Data Input' tool (middle mouse button) to see how this is achieved. We will work with this shot further.
2) (20%) Make a paper plot of the data (from '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 with 1-octave slopes at each end and a 1-octave bandpass (i.e., four numbers with frequency doubling each time; for example: 2-4-8-16). The two series need to overlap: often a series similar to the following is used
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 ~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, in the flow, enable the tool 'Bandpass filter' by right-clicking with the mouse. Middle-click, and in the parameters window, set the frequency parameters in the F1, F2, F3, F4 fields as specified above. Once they are set, press Ctrl-L to execute the flow and view the results.
4) (0%) Make hardcopies of the resulting panels - this is a set of filter test panels.
5) (5%) Which of these 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 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.