University of New Orleans ScholarWorks@UNO University of New Orleans Theses and Dissertations Dissertations and Theses Fall 12-15-2012 Calculation of a Synthetic Gather using the Aki-Richards Approximation to the Zoeppritz Equations Graham Ganssle [email protected] Follow this and additional works at: https://scholarworks.uno.edu/td Part of the Geophysics and Seismology Commons Recommended Citation Ganssle, Graham, "Calculation of a Synthetic Gather using the Aki-Richards Approximation to the Zoeppritz Equations" (2012). University of New Orleans Theses and Dissertations. 1566. https://scholarworks.uno.edu/td/1566 This Thesis is protected by copyright and/or related rights. It has been brought to you by ScholarWorks@UNO with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights- holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/or on the work itself. This Thesis has been accepted for inclusion in University of New Orleans Theses and Dissertations by an authorized administrator of ScholarWorks@UNO. For more information, please contact [email protected]. Calculation of a Synthetic Gather using the Aki-Richards Approximation to the Zoeppritz Equations A Thesis Submitted to the Graduate Faculty of the University of New Orleans in partial fulfillment of the requirements for the degree Master of Science In Applied Physics By Graham R. Ganssle B.S. University of New Orleans, 2009 December, 2012 Table of Contents List of Figures ............................................................................................................................................. iii Abstract ........................................................................................................................................................ iv Chapter 1: Introduction ................................................................................................................................ 1 Chapter 2: Background ................................................................................................................................. 3 Seismic Gathers ........................................................................................................................................ 3 Zoeppritz Equations and Linear Approximations .................................................................................... 5 Wireline Logging ....................................................................................................................................... 7 Chapter 3: Techniques and Methods ............................................................................................................ 9 Data ......................................................................................................................................................... 10 Wavelet Choice ....................................................................................................................................... 10 Normal Incidence Synthetic Trace ......................................................................................................... 11 Offset Incorporation and AVO ............................................................................................................... 13 Chapter 4: Analysis ...................................................................................................................................... 17 Algorithm Differences ............................................................................................................................ 18 1) Vertical Scale ............................................................................................................................... 18 2) Wavelets ..................................................................................................................................... 21 3) Sample Rates ............................................................................................................................... 22 4) Approximations ........................................................................................................................... 22 Comparison of the two synthetic gathers ............................................................................................. 23 Chapter 5: Conclusions and Future Work ................................................................................................... 27 Conclusions ............................................................................................................................................. 27 Future Work ............................................................................................................................................ 27 References .................................................................................................................................................. 28 Appendix ..................................................................................................................................................... 29 Vita .............................................................................................................................................................. 57 ii List of Figures Figure 1: Selected ray paths from source to receiver ................................................................................... 1 Figure 2: (left) wireline log data and (right) seismic data ............................................................................. 3 Figure 3: Seismic gather with offset angle in degrees on the horizontal axis and time in seconds on the vertical axis ................................................................................................................................................... 4 Figure 4: Graphical representation of an acoustic wave striking a boundary between two media with different acoustic impedances ...................................................................................................................... 5 Figure 5: Sonic logging tools. (left) short spaced, (right) long spaced .......................................................... 8 Figure 6: Map of southeast Louisiana with the well located at the tip of the red arrow ............................. 9 Figure 7: (left) Ricker wavelet r(t) with f = 1, (right) its Fourier transform R(ω) ........................................ 11 Figure 8: The normal trace plotted by using the ratio of reflected amplitude to incident amplitude for the reflection coefficient series of eq. 7 convolved with the Ricker wavelet of eq. 5. ..................................... 12 Figure 9: A zero offset trace plotted using the Aki-Richards equation ....................................................... 14 Figure 10: A synthetic gather for SEBB #39ST1 plotted using the authors algorithm. Horizontal scale is offset angle, with one trace for every five degrees. Vertical scale is data index number increasing downward ................................................................................................................................................... 16 Figure 11: A synthetic gather of SEBB #39ST1 plotted using IHS Kingdom ................................................ 17 Figure 12: A data index/depth map from the author's algorithm .............................................................. 19 Figure 13: The discrete derivative of the data in figure 12 ......................................................................... 19 Figure 14: A time/depth map from IHS Kingdom ....................................................................................... 20 Figure 15: The discrete derivative of the data in figure 14 ......................................................................... 21 Figure 16: (left) an Ormsby wavelet and (right) its Fourier transform ....................................................... 22 Figure 17: Synthetic gathers, (left) created using IHS Kingdom, (right) created using the author's algorithm ..................................................................................................................................................... 23 Figure 18: A cartoon of a class 1 AVO response (Hampson-Russell, 2005) ................................................ 25 Figure 19: A cartoon of a class 4 AVO response (Hampson-Russell, 2005) ................................................ 25 iii Abstract A synthetic seismic gather showing amplitude versus offset can be analyzed by the interpretive geophysicist to predict rock properties useful in oil exploration. For the research reported in this thesis, reflection coefficients derived from measured well log data are convolved with a Ricker wavelet to create a synthetic seismic trace. The Zoeppritz equations describe the propagation of an acoustic wave across an interface between two viscous media of different acoustic impedances with respect to increasing offset angle. The Aki-Richards linear approximation to the Zoeppritz equations is applied here to create a synthetic seismic gather with offset angles up to fifty degrees. The Aki-Richards approximation has not been used in this fashion prior to this research. The resulting gather is compared to a corresponding synthetic gather created using commercially available approximations and