GEOL 706 -- Fall 2010
LAB 2 -- Spectra and the Z-Plane

DUE Mon. 11 October 2010

The exercises in light gray color are extra credit and are not required.

Reading: Claerbout, 1992, Chapter 2 and Chapter 3.

  1. Claerbout, 1992, p. 14, exercise 4.

  2. Claerbout, 1992, p. 14, exercise 5.

  3. Download the jrg.jar Java archive file for the JRG Seismic Processing System. Run the new ZPlane tool from a Terminal window with the commands:
    	cd [folder where you put jrg.jar]
    	java -cp jrg.jar ZPlane &
    
    Read about the Zplane program in Appendix E of the text. In this new version, point at the Z-plane field to see the complex coordinates, omega, and rho of the place you are pointing at. To place a Zero, point where you want the zero to be and click the mouse. To place a Pole, hold down the Option key when you click. To remove a Pole or Zero, point at the root you want to remove, hold down the Shift key, and click the mouse on it. You can also point and type the Z or P key to place Zeroes and Poles, and point and type the X key to remove them. You can also pick up and drag roots. To put a root on an axis, you can point just outside the axis.

    Type L to load a data set, see its summed spectrum, and filter it. Download this PVIZplaneDataExample.zip ZIP archive, which contains the reflection record shown by the original Zplane program, as a JRG Pack and as SEG-Y. Unzip it. When you press L, you see a blank, zero data record. Click on it and press Command-L (Control-L on Windows) to load a JRG Pack. You can navigate to where you unzipped the archive, open the PVIZplaneDataExample folder that contains the JRG Pack, and double-click on any file inside. The data will open much as it appeared in the original Zplane interface. Select the example data set with the choice tool near the bottom of the ZPlane window.

    Turn in the answers to the following questions:

    1. Why does a pole on the real axis yield a sinusoidal function of time?
    2. Explain how a pole on the imaginary axis yields an exponential in . Why does ZPlane not let you put poles or zeros below the real axis?
    3. Explain why, for the approximate second derivative filter, the spectrum is close to the magnitude of .
    4. How do you make the second derivative of the Gaussian?
    5. At what range of time and offset is the lowest frequency of ``ground roll''?

  4. Claerbout, 1992, p. 50, exercise 3.

  5. Claerbout, 1992, p. 60, exercise 1.

  6. Claerbout, 1992, p. 64, exercise 1.

  7. Claerbout, 1992, p. 65, exercise 1.

  8. Claerbout, 1992, p. 72, exercise 2.

  9. Claerbout, 1992, p. 73, exercise 1. Develop the filter empirically using the Zplane program. Three terms of feedback means that there are three non-zero coefficients in the denominator. This sets a limit of the number of poles and zeroes you can use (as always with real filters there are conjugate poles and zeroes frequencies between 0 and the Nyquist). This notch2 figure from PVI shows you where to put the poles and zeroes for the notch filter. List the pole and zero locations in terms of rho and frequency, or complex coordinates. Extra credit for working out the feedback filter terms.

Some Possible Projects

These are listed to make you aware of possible applications of the ideas covered by this lab. Although 706 does not require a term project, you could use these for one of the two projects required in 757.

Deconvolve a single-channel seismic-reflection data set recorded in Lake Washington, that crosses the Seattle fault several times. Use both recorded source wavelets and spectral whitening in your trials.

Invert single-channel reflection data from Lake Washington for the reflectivity sequence, given data sets recorded at different ranges of frequency over the same location. Develop a transfer function between the reflectivity and other geophysical measurements of the profile.

Construct a minimum-phase multiple-notch filter for a data set contaminated by 60 Hz noise and harmonics. Filter the data set, and describe artifacts, confidence in the content of the filtered spectrum, and any phase changes in the data.

Implement a flexible time-varying, minimum-phase Butterworth filter that will accomplish band pass or reject operations on aritrarily-sized seismic data. Test it on a real data set.

Investigate the use of the slant-stack operator for estimating missing data. Implement a moving-window routine for computing local slantstacks, and test it by filling gaps in stacks, migrations, shot records, and images.

Investigate the properties of a tree-ring data set. Put quantitative bounds on the confidence of cross-correlations between overlapping data, and on the periodicities of climatic cycles.

Develop a method for computing a spectrum from unequally-spaced data. Explore the advantages and disadvantages of linear and spline interpolation, integrations, and model fitting. Speculate on 2-d spectra from randomly-spaced 2-d data.